Analytic and Meromorphic Functions

Appendix A Analytic and Meromorphic Functions

We list here some standard theorems for analytic and meromorphic functions from complex function theory without proofs. All functions are assumed to be single- valued. Curves (along which integrals are taken) are piecewise differentiable, and domains are connected (no disjunct pieces) and often even simply connected (no holes; see Sect. 3.10 for details). Exchanging sum and integral ∞ Aseries f j with f j : D → C converges uniformly in B ⊂ D if j=0 ∞ ∀>0 ∃N() ∈ N ∀n ≥ N ∀z ∈ B : f j (z) <. (A.1) j=n

∞ Note that N does not depend on z.LetC be a curve and f j a function series. Let j=0 f j converge uniformly on C to a function f deﬁned on C. (We identify the curve C—a map—with its set of image points.) Then dzfj (z) = dz f j (z) = dzf(z). (A.2) C C C j j

Proof: Remmert (1992), p. 142, Bieberbach (1921), p. 111. The Goursat lemma Let the function f be analytic in the domain D. Consider a triangle whose boundary and interior lie completely in D.LetC be the boundary of this triangle in D. Then dzf(z) = 0. (A.3) C

© Springer Nature Switzerland AG 2019 535 A. Feldmeier, Theoretical Fluid Dynamics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-31022-6 536 Appendix A: Analytic and Meromorphic Functions

Lemma and proof are classic, see Remmert, p. 148, Bieberbach, p. 118. The Cauchy theorem A domain D is star-shaped if there exists an interior point a such that for any point b ∈ D, the straight line from a to b lies in D.Let f be analytic in the star-shaped domain D. Then for each closed curve C in D (not touching its boundary), dzf(z) = 0. (A.4) C

Thus the line integral from one point to another depends only on these points, not on the path taken. The theorem also holds if D is not star-shaped, but simply connected. For the proof, see Remmert p. 151, Bieberbach, p. 115, and Ahlfors (1966), pp. 139– 143. C is called homologous to 0 if C does not enclose points outside D. This means, C does not go round holes in D. For an exact deﬁnition using winding numbers,see (A.22). Cauchy’s theorem holds if f is analytic in a (multiply connected) domain D, and if C is a cycle that is homologous to 0 in D. Proof: Ahlfors (1966), pp. 144–145, Dixon (1971). The Cauchy integral formula Let f be analytic in a domain D.LetB be the set of points within a circle ∂ B in the complex plane. (In R3, B is a ball; in the complex plane, B is a disk.) Let B ∪ ∂ B ⊂ D. Then for all points z in B (not on ∂ B), 1 f (ζ ) f (z) = dζ (A.5) 2πi ∂ B ζ − z

Thus, any f analytic in a disk is fully determined by f on the boundary circle of the disk. For the proof, see Remmert p. 159 or Bieberbach, p. 123 and p. 128. Since Cauchy’s formula is so fundamental, we give a simple derivation. (The modern proof f (ζ )− f (z) ( ) is more elegant, using ζ −z and f z .) Let be a circle with radius r and center z. For points on ,

ζ = z + reiϕ, ∂ζ ζ = ϕ = ireiϕ ϕ, d ∂ϕ d d (A.6) giving dζ 2π = i dϕ = 2πi. (A.7) ζ − z 0

Since f is continuous, one has for r → , with 1, dζ f (ζ ) f (z) = dζ (A.8) ζ − z ζ − z Appendix A: Analytic and Meromorphic Functions 537

(for small , lies completely in B), or f (ζ ) 2πif(z) = dζ . (A.9) ζ − z

Let K be an arbitrary curve that starts on ∂ B and ends on . Then the curve = K − − K + ∂ B deﬁnes a closed cycle. The signs refer to the direction in which the segments are passed. (Regarding K and −K , one can think of two banks of one curve or of two curves that have inﬁnitesimal distance everywhere.) f is analytic in the simply connected domain inside . From Cauchy’s theorem, with contributions from K and −K cancelling, f (ζ ) f (ζ ) dζ − dζ = 0, (A.10) ∂ B ζ − z ζ − z and using (A.9), Cauchy’s formula (A.5) follows. The Morera theorem In Cauchy’s theorem, ‘ f is analytic’ implies dzf(z) = 0. Morera’s theorem gives conditions for the implication in the opposite direction. Let f be continuous in a ( ) = domain D.Let C dzfz 0 for all closed curves C in D. Then f is analytic in D. Proof: Remmert p. 185, Bieberbach, p. 133. Normal convergence ∞ Aseries f j of functions f j : D → C converges normally in D if every point in j=0 D has a neighborhood U such that | f j |U < ∞. Theorems on normal convergence = If all f j in f j f are continuous and converge normally, f is continuous. Let τ : N → N be one-to-one. If f j is normally convergent, then fτ(j) is. The limit is in both cases the same function f . Proof: Remmert, p. 82. The Cauchy–Taylor theorem Let B be the interior of a circle with center c, and f be analytic in B. Then there is ∞ j aseries a j (z − c) that converges normally to f in B.Thea j are given by, with j=0 C a circle that lies wholly within B, 1 f (ζ ) a j = dζ . (A.11) 2πi C (ζ − c) j+1

Furthermore, the Taylor formula holds, 1 d j a = f (z) . (A.12) j ! j j dz z=c 538 Appendix A: Analytic and Meromorphic Functions

Thus, every analytic function has a Taylor series expansion,

∞ j f (z) = a j (z − c) , (A.13) j=0 where c is an arbitrary but fixed point in the analytic domain. Proof: Remmert, p. 165, Bieberbach, p. 135. The Riemann continuation theorem Let D be a domain, and c ∈ D an arbitrary point. Let f be analytic in D\c (short for D\{c}) and continuous in D. Then f is analytic in D. Proof: Remmert, p. 167, Bieberbach, p. 145. Infinite differentiability Let f be analytic in a domain D. Then f is arbitrarily often complex differentiable in D. Thus every analytic function can be differentiated infinitely often. This is quite clear from the Cauchy–Taylor theorem above. Proof: Remmert p. 169, Bieberbach, p. 132. Identity theorem. I Let f and g be analytic in the interior B of a circle. Let f |U = g|U for an arbitrarily small neighborhood U ⊂ B. Then f = g in B. The reason is that the Taylor coefficients can be calculated in U and held in B. Identity theorem. II

Let f and g be two functions analytic in a domain D. Let the set of points c j where f (c j ) = g(c j ) have a limit point b in D. Then f = g in D. For the proof, see Remmert, p. 179 or Bieberbach, p. 137. The proof idea is as follows. At b,all derivatives of f and g agree. By the Cauchy–Taylor theorem, f and g are then identical in D. Double series theorem (Weierstrass) ∞ Let all f j (z) with j = 0, 1, 2,...be analytic in a domain D.Let f (z) = f j (z) j=0 = converge uniformly. Then f is analytic, and f f j . Proof: Remmert, p. 196, Bieberbach, p. 153. The Vitali theorem

This is an extension of the foregoing theorem. Let D be a domain and C ={c j }⊂D m a point set with limit point in D. We assume ∃M ∀m ∀z ∈ D : f j (z)

Discrete zeros Let f = 0 be analytic in a domain D. Then the set of zeros of f is discrete: for any two zeros there exist nonoverlapping neighborhoods. Proof: Remmert, p. 182, Bieberbach, p. 138. Conformal mapping f : D → C is called conformal if it preserves angles locally. If two curves cross at angle α, their image curves also cross at angle α.Let f be analytic in a domain D, and nonconstant in every neighborhood. Then there exists a discrete set A ⊂ D so that f is conformal in D\A.ThesetA consists of the points where f = 0. At these points, f is not conformal. Proof: Remmert, p. 187, Bieberbach, p. 41. Liouville’s theorem Let f be analytic and bounded in the whole complex plane. This means there is some M > 0 such that | f (z)|≤M for all z ∈ C. (A.14)

Then f is constant on C. Proof: Remmert, p. 192, Bieberbach, p. 150. Maximum property or maximum principle Let f be analytic in D and let | f | have a local maximum in D. Then f is constant in D (compare with Sect.3.4). Equivalently, if ∂ D is the boundary of D, and D¯ = D ∪ ∂ D, then | f (z)|≤|f (ζ )| for all z ∈ D¯ and ζ ∈ ∂ D. (A.15)

Proof: Remmert, p. 191 and p. 203, Bieberbach, p. 143. Open mapping theorem If f is analytic in D and not constant in D, then f (G) is a domain. Thus, analytic functions map open sets U to open sets f (U). Proof: Remmert, p. 202, Bieberbach, p. 187. The general invariance of domain theorem for two dimensions is as follows: If U ⊂ R2 is open and f : U → R2 is injective and continuous, then f (U) is open. The proof is by Schoenﬂies (1899), with simpliﬁcations by Osgood (1900) and Bernstein (1900), and can be found in Markushevich (1965), p. 95. The proof of the invariance of domain theorem for general Rn is by Brouwer (1912). Lemma of Schwarz If f : E → E, with unit disk E ={z :|z| < 1}, and if f (0) = 0, then

| f (z)|≤|z| for all z ∈ E. (A.16)

Proof: Remmert, p. 212. 540 Appendix A: Analytic and Meromorphic Functions

Study theorem

−1 Let both f : E → D (with unit disk E) and f be analytic. Let Dr be the image under f of a circle with radius 0 < r < 1. Then if D is convex, so are all Dr . Proof: Remmert, p. 215, using the lemma of Schwarz.

We now allow f to have isolated singularities where f is not analytic. Isolated means that for any two singularities, disjunct neighborhoods exist. Singularities are not generally isolated, e.g., there exist continuous curves of singularities. For exam- ∞ ple, for the function deﬁned by the series z2 j , each point on the convergence cir- j=1 cle |z|=1 is singular. This so-called Weierstrass series is discussed in Bieberbach, p. 213. A function ∞ j f (z) = a j (z − c) (A.17) j=−m with a−m = 0 has a pole of order m at c. The series (A.17) with m ≥ 1istermeda Laurent series.Ifm ≡∞, then f is said to have an essential singularity at c.If f has a pole at z = c, then 1/f is bounded in some neighborhood of c.Thisisnot the case if c is an essential singularity. At poles, one often considers 1/f instead of f . Poles are isolated A limit point of poles is an essential singularity. This is a direct consequence of Liouville’s theorem. Proof: Bieberbach, p. 150. Meromorphic and rational functions A complex function f is meromorphic if f is analytic in D except at isolated poles. A rational function is the quotient of two polynomials in z. Poles and rational functions If f is meromorphic in the whole of C, then f is a rational function. Proof: Bieber- bach, p. 151, Neumann (1884), p. 60. The Casorati–Weierstrass theorem Let c be an isolated, essential singularity, and U any neighborhood of c. Then f (U\c) is dense in C. More speciﬁcally, for any >0 and any b ∈ C, there is an a ∈ U so that | f (a) − b| <, i.e., f (a) comes close to b. The case f (a) = b is subject of Picard’s theorem. Proof: Remmert, p. 242, Bieberbach, p. 149. Identity theorem. III Let f and g be two meromorphic functions in a domain D. The set of points where f (c j ) = g(c j )<∞ shall have a limit point. Then f = g in D.Proof:Remmert, p. 252, Bieberbach, p. 137. Appendix A: Analytic and Meromorphic Functions 541

The Cauchy theorem for rings A ring R is a complex domain with an inner and outer circular boundary. A ring is twofold connected. Let f be analytic in the ring. Let the circles C1, C2 belong to the interior of the ring. Then dzf(z) = dzf(z). (A.18) C1 C2

Proof: Remmert, p. 273, Bieberbach, p. 127 and p. 129. The Cauchy integral formula for rings The boundary ∂ R of a ring consists of two circles. Integration proceeds in opposite directions along these circles. Let f be analytic in D, and R bearingsothatR ∪ ∂ R ⊂ ∈ D. Then for z R, 1 f (ζ ) f (z) = dζ . (A.19) 2πi ∂ R ζ − z

Proof: Remmert, p. 274, Bieberbach, p. 139 and p. 145. The Laurent series A direct consequence of this formula is the Laurent series theorem. Let f be analytic in the ring R with center c. Then there is a unique Laurent series that converges normally to f , ∞ j f (z) = a j (z − c) . (A.20) j=−∞

For any circle C inside the ring (not its boundary), the a j are given by 1 f (ζ ) a j = dζ . (A.21) 2πi C (ζ − z) j+1

Proof: Remmert, p. 276, Bieberbach, p. 139. Winding number Let C be a closed curve and z a point not lying on C. Then 1 dζ indC (z) = ∈ Z (A.22) 2πi C ζ − z is the winding number (number of turns) of C around z.Theinterior of C is the set of all points with ind = 0. The exterior of C is the set of all points with ind = 0. A curve is simple closed,ifind= 1 for all points in its interior. 542 Appendix A: Analytic and Meromorphic Functions

Null-homolog A curve C is null-homolog in a domain D, if its interior lies in D. Alternatively, C is null-homolog if it does not go round points of the complement of D. That is, a curve is null-homolog if indC (z) = 0 for all z ∈ C\D. Residue Let C be a circle with center c and inﬁnitesimal radius. Let f be analytic except at isolated poles or essential singularities. The residue Resc f of f with respect to c is the integral 1 Resc f = dzf(z). (A.23) 2πi C

If f is analytic at c, then Resc f = 0 according to Cauchy’s theorem. Let ∞ j f (z) = a j (z − c) (A.24) j=−∞ for f analytic in a neighborhood U of D\c. Then 1 Resc f = dzf(z) = a−1, (A.25) 2πi C where the circle C lies in U. For the residue at the point c →∞, see Bieberbach, p. 172 and p. 174. We have now immediately (see Remmert, p. 304, Bieberbach, p. 197): if c is a pole of f of ﬁrst order, then

Resc f = lim (z − c) f (z). (A.26) z→c

If c is a pole of f of order m, then

m−1 1 d m Resc f = lim (z − c) f (z) . (A.27) (m − 1)! z→c dzm−1

For essential singularities, there is no simple formula for the residue. Residue theorem

Let C be a closed, null-homolog curve in a domain D.Let{c j } be a ﬁnite set of n points, where number of the c j lies on C.Let f be analytic in D\{c j }. Then

n 1 dzf(z) = indC (c j ) Resc f. (A.28) 2πi C j j=1

For the proof, see Remmert, p. 306 or Bieberbach, p. 174. In most applications, C is chosen so that indC (c j ) = 1 for all c j :ifC is simple closed and null-homolog in D, Appendix A: Analytic and Meromorphic Functions 543 then n 1 dzf(z) = Resc f (A.29) 2πi C j j=1

There are many applications of this theorem; we mention one. Let f be analytic on the whole real axis, except possibly at inﬁnity. Let f be analytic in the upper half-plane except at singularities c1 to cn. Finally, f shall vanish more strongly than 1/z for large z,

∀>0 ∃r > 0 ∀z :|z| > r →|zf(z)| <. (A.30)

Then the integral of f along the real axis converges and is given by

∞ n dzf(z) = 2πi Res f. (A.31) −∞ c j j=1

Proof: Bieberbach, p. 175. The Rouche theorem Let f and g be analytic in D.LetC be a simple closed and null-homolog curve. For all points z on C,let | f (z) − g(z)| < |g(z)|. (A.32)

This also means that g(z) = 0onC. Then f and g have the same number of zeros in the interior of C. Proof: Remmert, p. 310, Bieberbach, p. 185. Monodromy theorem

Let D be an open disk with center z, and f0 : D → C analytic (see Fig. A.1). Let C and C be two homotopic curves from z to z , and D an open disk with center z .If f, f : D → C are analytic continuations of f0 along C and C , then f = f .Note that general curves C and C can only be homotopic if they lie in a simply connected domain. Proof: Jänich (1993), p. 59 and Markushevich (1967), p. 269.

C f 0 f 0 z z 0 f 0 0 D C D

Fig. A.1 Analytic continuation of the function element ( f0, D) centered at z to the function element ( f, D ) centered at z along the curve C, and to the function element ( f , D ) along the homotopic curve C 544 Appendix A: Analytic and Meromorphic Functions

References

Ahlfors, L.V. 1966. Complex analysis, 2nd ed. New York: McGraw-Hill. Bernstein, F. 1900. Ueber einen Schönflies’schen Satz der Theorie der stetigen Funktionen zweier reeller Veränderlichen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1900: 98. Bieberbach, L. 1921. Lehrbuch der Funktionentheorie, Band I, Elemente der Funktionentheorie. Leipzig: B.G. Teubner. Wiesbaden: Springer Fachmedien. Brouwer, L.E.J. 1912. Beweis der Invarianz des n-dimensionalen Gebiets. Mathematische Annalen 71: 305 and 72: 55. Dixon, J.D. 1971. A brief proof of Cauchy’s integral theorem. Proceedings of the American Math- ematical Society 29: 625. Jänich, K. 1993. Funktionentheorie. Berlin: Springer. Markushevich, A.I. 1965. Theory of functions of a complex variable, vol. I. Englewood Cliffs: Prentice-Hall. Markushevich, A.I. 1967. Theory of functions of a complex variable, vol. III. Englewood Cliffs: Prentice-Hall. Neumann, C. 1884. Vorlesungen über Riemann’s Theorie der Abel’schen Integrale, 2nd ed. Leipzig: Teubner. Osgood, W.F. 1900. Ueber einen Satz des Herrn Schönflies aus der Theorie der Functionen zweier reeller Veränderlichen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1900: 94. Remmert, R. 1992. Funktionentheorie 1, 3rd ed. Berlin: Springer. Schoenflies, A. 1899. Ueber einen Satz der Analysis Situs, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse 1899: 282. Index

A Akers, B.F., xi, 415, 416 Abbott, D.C., 286 Alabama, 207, 208 Abernathy, F.H., ix, 224 Albina-belenkaya, 487 Abramowitz, M., 156, 288, 289, 348, 349, Aldersey-Williams, H., 304, 308 353–355, 360 Algebraic function, 113 Absorption, 275, 284–286, 290–292, 458, degree of, 113 466 Algorithm, 402 coefﬁcient, 285 Almost all, 63 shadow, 286 Aluminium foil, 473, 475 Accretion, 153 Amick, C.J., 399 disk, 2, 115, 152–154, 157 Amphidromic point, 308–312, 380 Acoustic cutoff, 460, 462–465, 468–470 Amplitude front, 380 Acoustics, 346, 457 Analytic Action, 54, 212, 213 continuation, 186, 187, 290, 292, 358, Adhesion, 362 362, 401, 404–406, 543 Adiabatic, 458, 465–467, 486 continuation along a curve, 405 curve, 486 function, 79, 96–101, 108–110, 116, 118, energy equation, 467, 486, 502 128, 133, 165, 166, 182, 186, 187, exponent, 458, 459, 465, 480, 482, 483, 191, 198, 201, 203, 234, 235, 246, 503, 504, 514, 516, 517, 526–528 249, 290, 352, 353, 355–359, 367, 397, process, 216, 465, 494, 502, 505, 512, 403, 405–407, 409, 427, 442, 449, 507, 514, 516, 526, 527 535–543 sound speed, 465 norm, 249 Adjacent Angular column, 464 momentum, 8, 152–154 crest, 281 speed, 31, 152, 153, 207, 276, 305, 341, parcel, 18, 19 342, 346, 470, 471 state, 504, 525, 526, 531 Antenna mast, 136 vortex, 126, 142, 221, 234 Arclength, 45, 48, 49, 86, 232, 261, 264, 266, Advection, 44, 154, 250, 265, 277, 314, 507, 349, 410, 514, 517, 521, 523, 527 513, 514 Arctic, 308 Ahlfors, L.V., 98, 174, 186, 358, 405, 536 Aref, H., 159 Air, 136, 161, 163, 215, 277, 394, 414, 457, Arnold, V.I., 454 458, 465–467, 473, 475, 477 Associativity, 8, 65 foil, 134, 192 Astrophysics, 152, 161, 162, 474 plane, 130, 134 Asymptotic, 161, 178, 238, 279, 280, 361, Airy, G.B., 309, 315, 383, 384 362, 402 © Springer Nature Switzerland AG 2019 545 A. Feldmeier, Theoretical Fluid Dynamics, Theoretical and Mathematical Physics, https://doi.org/10.1007/978-3-030-31022-6 546 Index

analysis, 239, 240 Bessel, F.W., 156–158, 289, 291, 332, 348, Atlantic, 136, 295, 308, 309, 346 354, 360, 428 Atmosphere, 209, 216, 255–257, 457, Bessis, D., 507, 508, 512 460–467, 469, 489, 491, 502 Betchov, R., 263, 264 Atom or molecule, 1, 6, 17, 24, 28, 276, 458, Betz, A., 2, 107 475, 480, 482, 527 Bieberbach, L., 109, 110, 183, 184, 535–543 Auld, D., ix, 474, 475 Big Bubble Curtain, 395 Average, averaging, 6, 61, 62, 84, 480 Binomial theorem, 245 Axford, W.I., 502 Binormal, 262 Axler, S., 84 Biot, J.B., 48 Biot–Savart law, 48 Biretta, J.A., 162 B Birkhoff, G.B., 161, 164, 172, 193, 222–224, Back- 230, 233, 247, 249, 250 scattering, 291, 292 Birkhoff–Rott equation, 230, 233, 247, 250 wash, 334, 341 Birmingham, 208 Baja California, 377 Biuniquely, 174 Baker, G.R., ix, 34, 224, 234, 250, 251, 255 Bjerknes Balance law, 8, 154, 291, 394, 477–480 theorem, 213, 215, 216 Banach space, 419, 422, 423, 433 tube, 214, 215 Banach, S., 419, 422, 423, 433 Bjerknes, V., 213–216 Bank (of curve), 108, 111, 112, 174, 184, Blaschke, W., 262 537 Blasius theorem, 130, 131 Barcilon, V., 363 Blasius, P., 130, 131 Barents Blast, 323 Island, 307 Blood, 207 Sea, 307 Boltzmann Barometric law, 255, 256, 460–462 collision term, 28, 29 Barotropic, 6, 43, 48, 209, 213, 266, 459, constant, 6, 254, 480 494, 516 Boltzmann, L., 6, 15, 24, 28, 29, 254, 480 Barrier, 85, 346, 347, 349–352, 359, 361, 362 Borda, J.-C., 161, 171 Basin, 310, 311, 315 Bore, 3, 205, 295, 299, 313, 333–336, Basye, R.E., 98 338–340 Batchelor, G.K., 24, 31, 37, 40, 61, 163, 210, adjustment, 334, 335 212, 216, 217, 221 at shore, 334, 340, 341 Bateman, H., 60 heat generation, 336, 338 Ben-Dor, G., 477 jump, 334–336, 338, 340, 341, 396 Benedetto, D., 35 layer, 335 Bernoulli equation, 48, 49, 56, 59, 131, 161, pre/post region, 335–340 164, 178, 368, 371, 372, 386, 398, primary, 341 402–404, 406, 407, 411, 421, 447, propagation, 338 448 rest frame, 335, 338 Bernoulli, D., 48–50, 56, 131, 161, 164, secondary, 341 178, 368, 371, 372, 386, 397, 398, speed, 334–336, 338–340, 521 402–404, 406, 407, 411, 421, 447, transition, 334, 336, 338, 339, 396, 401, 448 521 Bernstein theorem, 246 turbulence, 338 Bernstein, F., 539 Borehole, 363 Bernstein, S.N., 246 Bott, R., 19, 66, 71, 73, 75 Bessel Boundary differential equation, 156, 157, 332, 354 condition, 12, 28, 46, 47, 50, 56, 57, function, 156, 158, 289, 291, 348, 360 59, 79, 85, 89, 90, 95, 118–120, 123, inequality, 428 124, 129, 133, 148–150, 156, 163–165, Index 547

192, 199, 201, 203, 204, 242, 243, 258, C 259, 297, 318–323, 349–351, 362, 363, Caﬂisch, R.E., 239, 246, 248, 249, 509, 510 367–369, 372, 386, 389, 397–399, 402, Caltech thesis library, ix 412–414, 419–422, 441–443, 447 Canada, 335 condition, Dirichlet, 82, 83, 85, 87, 124, Canal, 147, 257, 258, 295–297, 300, 303, 362, 363, 412 304, 306, 307, 311, 312, 314, 324, condition, Neumann, 82, 83, 90, 91, 124, 335, 362, 368, 371, 374, 376, 385, 133, 362, 363, 411, 443 394 condition, nonreﬂecting, 321 Cannon, C.J., 491, 492 condition, no-slip, 362, 363 Canonical condition, perfect slip, 132, 362, 363 coordinates, 14, 15, 51 layer, 28, 260 momenta, 14, 15, 51 value problem, 88, 89, 119, 161, 227, variables, 14, 15, 51, 55, 60, 252 367, 402 Cape Bessel, 307 Bound-bound transition, 284, 285, 291, 292 Capillarity, 401, 408, 409, 414–416, 419, Bounded, 422, 433, 437, 452 457 distance, 145, 224 Carrier, G.F., 328, 333 function, 9, 350, 352, 353, 355, 356, 358, Cars, 477 359, 437, 539, 540 Cartan, É., 63 set, 12 Cartan, H., 100 Boyle–Mariotte law, 458 Cartwright, D.E., 311 Boyle, R., 458 Case distinction, 360, 476 Brachistochrone, 56 Case, K.M., 28, 256 Branch, 101, 108, 469 Casorati, F., 540 point, 109, 110, 113, 195–197, 243–245, Casorati–Weierstrass theorem, 540 249, 401, 407, 507, 510, 511 Cauchy point order, 109, 511 inequality, 453 Brasher, P., 394 integral formula, 131, 359, 536, 537, 541 Breaker, see Wave breaking theorem, 361, 536, 537, 541, 542 Breeze solution, 491, 492 Cauchy, A.-L., 18, 24, 81, 96–98, 116, 131, Brennen, C.E., ix, 163, 172 133, 165, 168, 199, 203, 204, 232, 233, 359, 361, 371, 409, 422, 442, Bretherton, F.P., 213, 270 443, 445, 448, 453, 536–538, 541, Bridging law, 286, 287, 289 542 Brillouin principle, 192 Cauchy–Riemann equations, 2, 81, 96–98, Brillouin, M., 191, 192 116, 133, 165, 168, 203, 204, 409, Britain, 295, 308, 335, 346, 362 422, 442, 443, 448 Broer, L.J.F., 60 Cauchy–Taylor theorem, 537, 538 Brouwer, L.E.J., 539 Causality, 517 Brown, G.L., 255 Cavitation, 161 Brunt, D., 464, 465, 468, 469 Cavity, 79, 100, 113, 161–163, 172, 178, Brunt–Väisäläfrequency, 464–466, 468–470 179, 181, 184, 185, 191–195, 199, Bubble 367, 401, 403, 406, 409, 414, 470 breakwater, 394, 395 Cell, 39, 531 curtain, 394, 395 Centre vapor or air, 163, 323, 394, 395, 414 d’études acadiennes, ix, 335 Buffoni, B., 420 of mass, 394, 458 Burgers Centrifugal acceleration, 39, 49, 275 equation, 113, 507–510, 512 Chamberlain, J.W., 491 turbulence, 512 Chambers, Ll.G., 362 Burgers, J.M., 113, 507–510, 512 Chandrasekhar, S., 226, 228, 229, 256, 467 Burke, E.R., 224 Chaos, 291 Burkhardt, H., 92 Characteristic 548 Index

crossing, 514, 518, 519, 521, 524, 525 speed potential, see Potential, complex curve, 284, 477, 509, 510, 512–514, Compressibility, compression, 9, 24, 25, 516–519, 521–531 457, 458, 464, 473, 479, 486, 514 value, 423, 427, 428, 432, 440, 443, 448, Compressor, 394 451 Computational fluid dynamics, 531 Chimney, 136, 138 Computer, 221, 225 Chorin, A.J., 35, 43, 130, 212, 512, 513 Concave, 192 Christoffel, E.B., 42, 79, 100–102, 104, 105, Configuration space, 253 107, 164, 166–169, 171, 174, 178, Confluent hypergeometric functions, 256 198 Connected Christoffel symbol, 42 multiply, 536 Chromosphere, 491 n-fold, 174 Circle theorem, 133 Riemann sheets, 109, 112, 113 Circulation, 33, 34, 36, 115–117, 126, set, 5, 12, 34, 65, 84, 85, 98, 535 130, 131, 133, 134, 140, 189, 207, simply, 13, 14, 64, 86, 98–100, 166, 367, 209–216, 231–236, 247, 249, 259, 405, 445, 535–537, 543 264, 266–268 twofold, 541 absolute, 216 Conservation law, 8, 10, 144, 154, 213, 216, theorem, 33, 207, 209–213, 266 323, 334, 335, 337, 392, 476–478, Cisotti, U., 192 484, 485 Classical mechanics, 454 Conserved Clayden, W.A., 194, 195, 197, 199 angular momentum, 213 Clebsch, A., 50, 51, 53, 56, 57, 270 circulation, 266 Closed set, 84, 85 crest number, 281 Closure, 12, 90, 186 energy, 8, 48, 213, 462, 476, 477, 479, condition, 189 480, 484 Cloud, 9, 14, 136, 152, 208, 224, 250 entropy, 209, 216, 458, 467, 473, 476, Coast, 295, 329, 341, 343, 344, 346, 377, 478, 486, 492, 494, 502, 503, 505, 512 378, 380–383, 385, 394 interaction energy, 224 Coastal strip, 341, 346 mass, 8, 9, 11, 335, 336, 476–479, 484 Cocke, W.J., 17, 61 momentum, 8, 9, 142, 213, 335, 476, 479, Coefficient, 41, 42, 62, 89, 90, 96, 151, 158, 484 191, 241, 246, 247, 407, 410, 493, point number, 15 508, 515, 519–521, 538 vorticity, 45 Coherent, 251, 252, 255, 276, 385 wave energy, 336 Cohomology, 73, 75 Consistent, consistency, 28, 70 Cole, J.D., 512 Constant Collective excitation, 276 of motion, 10, 11, 14, 15, 55, 61, 212, Collinear atoms, 458 213, 223, 224, 265, 269, 380 Collisionless state, 324, 326, 478, 524–527, 529–531 plasma, 276 Constraint, 50, 146 system, 29 Continuity equation, 9, 11, 29–31, 44, 153, Collision of atoms, 28, 29, 153, 276, 292, 296, 298, 300, 305, 312–314, 317, 491 318, 320, 324, 338, 343, 344, 386, Compact 459, 461, 467, 487–489, 502, 514, domain or set, 8, 34, 85 516 operator, see Operator, compact Continuously differentiable, 6, 35, 63, 88, Compass, 344 90, 257, 276, 283, 314, 334, 349, 519, Complex 521, 523, 524 scalar product, 423 Continuum, 12, 24, 98 speed, 79, 96, 130–132, 137, 139, 140, Boltzmann, 24, 154 167, 171, 174, 182, 183, 188–190, Cosserat, 24, 154 193–195, 197, 200, 231, 399, 421 limit, 15 Index 549

Contractible, 63–65, 74, 75, 80, 90, 91 Current Contraction, 9, 13, 14, 74, 99, 101, 152, 153, back-, 215 171, 323 density, 48 Convection, 467, 468, 470 electric, 35, 268 Convergent, 89, 98, 247, 267, 317, 373, 402, sheet, 35, 36 420, 425, 434–436, 438, 439, 447, Curvature, 264, 444 448, 450, 452, 453, 535, 537, 538, and torsion, 261, 263–265, 279, 368 540, 541, 543 continuous, 232 Convex Domain, 47, 192, 540 radius, 45, 49, 261, 264, 410 Convexity condition, 486 singularity, 36, 230, 232, 234, 243, 244, Convolution, 291, 351, 423 246, 248–251, 401, 507 theorem, 287, 352, 360 terms, 39 Cooling, 215, 505, 507 Curve net, 93, 515 Coordinate transformation, 70, 71, 128, 165, Curvilinear coordinates, 39, 45, 515 515 Cusp, 127, 184, 185, 188, 191, 192, 234, 248, Copson, E.T., 96, 198, 280, 351, 354, 361 395–402, 404, 408, 521 Co-range line, 308 Cut, 108, 109, 249, 401, 501, 511 Coriolis acceleration, 32, 39, 309, 341, Cycle, 99, 109, 536, 537 343–346, 353, 361 Cylinder (in a flow), 115, 127, 132–135 Coriolis, G.G.de, 32, 39, 309, 341, 343–346, Cylindrical coordinates, 8, 37, 38, 42, 259 353, 361, 362 Corner flow, 115, 118–121, 123, 126, 127, 147, 152, 161, 163–165 D Corona, 32, 36, 489, 491, 492, 502 D’Alembert, J., 6, 96, 132, 172 Corotating frame, 32 Dam break, 313, 323, 326–328 Cosserat, E. and F., 24, 154 Darwin, C., ix, 309 Cotidal line, 308–312 Darwin, G.H., 308, 309 Coulomb, C.A.de, 48, 230, 276 Davis, A.M.J., 122, 127 Coulomb gauge, 48, 230 Day, 305, 342 Countably infinite, 407 Dean, W.R., 125 Counterclockwise, 94, 98, 216, 217, 221, Decay 224, 252, 341, 346 exponential, 156, 246, 346, 370, 390, 469 Couple stress, 20, 22, 24, 154 Fourier coefficient, 246, 247 Coupling of distant locations, 284 perturbation, 256 Courant, R., 88–90, 113, 174, 232, 233, 477, rate, 157 479, 486, 502, 504, 512–515, 517, Decomposition, 19, 27, 28, 51, 90 519, 521, 523–526, 528–531 De-excitation, 291, 292 Cowley, S.J., 249, 250 Degrees of freedom, 14, 154, 207, 254, 458, Cox, A.D., 194, 195, 197, 199 480 Crack, 473 De l’Hospital, 453 Crapper, G.D., 297, 368, 401, 408, 409, 412, Denmark, 362 414, 415, 419 Dense Crease, J., 295, 299, 346, 350, 351, 354, 355, set, 540 357–359, 361, 362 shell, 504 Crest length, 380, 381, 383, 385 De Rham cohomology, 75 Crew, S.C., 402, 406, 407 Derivative in direction or along curve, 202, Crighton, D.G., 362 318, 494, 514–516, 519, 521–523, Critical 528, 530 angle, 125 Descartes, R., 7, 8, 11, 12, 14–16, 18, 19, 21, point, 323 23–27, 29, 30, 41, 51, 61, 100, 118, time, 225, 234, 241, 246–248, 250, 507 147, 191, 210, 211, 218, 225, 230, value, 125 283, 296, 341, 360, 385, 399, 467, Curle, N., 256 484 550 Index

Deshadowing, 285 relation, 3, 207, 228, 265, 276, 277, 279, Detach(ment), 64, 161, 163, 166, 184–186, 281, 282, 286, 292, 295, 344–347, 368, 194, 199, 260, 404 377, 378, 384, 408, 462, 463, 467–469, Determinant, 61, 124, 143, 146, 325, 427, 471 433, 467, 517, 522, 524 Displacement, 140, 300, 305–307, 342, 370, inﬁnite, 427, 431 465 Deviation (from mean), 391 Dissipation, 12, 395, 462, 491, 501 Diagnostic diagram, 469 Dissociation, 458 Diagonalization, 25, 27, 61 Distribution function, 6, 29, 62, 491 Diaphragm, 473, 475 Divergence-free, 2, 10, 11, 14, 15, 35, 36, Diatomic gas, 24, 458 46, 50, 51, 56, 61, 79, 80, 90–94, 98, Dieudonné, J., 433 115, 132, 134, 147, 164, 226, 266, 296, 385, 386, 395, 402, 409, 420 Difference scheme, see Finite difference Divergent, 89, 157, 223, 374 scheme Dixon, J.D., 536 Differentiability, 6, 453, 475, 477, 538 Dolaptschiew, Bl., 143 Differential form, 63, 65, 66, 68–70, 72, 74, Domain, 12, 63–65, 68, 71, 81, 82, 84, 85, 75 88, 90, 91, 98–100, 116, 166, 168, closed, 63, 65, 73, 75 172–174, 178, 186, 195, 200, 223, exact, 63, 65, 73, 75 224, 356, 358, 367, 401, 403–406, Differential geometry, 212 445, 518, 519, 521, 525, 535–543 Diffraction, 346, 361 of dependence, 518 Diffusion, 156 Domm, U., 141, 143, 144, 146 equation, 156 Doodson, A.T., x, 309 Dilation, 15, 83 Doppler, C.A., 285, 286 Dimension, 68 Doppler effect, 285, 286 Dimensionless, 323, 325, 376 Double Dirichlet periodic, 197 integral, 88, 203 ratio, 183 principle, 88–90, 92, 203 shock, see Shock pair Dirichlet, P.G., 82, 83, 85, 87–90, 92, 124, Downstream, 252, 292, 501 203, 362, 363, 412 Drazin, P.G., 216, 255–257 Discontinuity, 35, 36, 110, 161, 218, 225, Drift, 153, 159, 295 231, 233, 234, 250, 257, 282–284, Duncan, J.H., 399 295, 328, 334–336, 338, 340, 341, Dunford, N., 424, 433 350, 359, 360, 363, 395, 401, 404, Dussan, V., 12 444, 445, 470, 477, 482, 485, 486, Dust, 152, 153 504, 519, 521 devil, 117 contact, 504–507, 526 Dyachenko, S.A., 402, 406 speed of, 328, 334, 335, 338, 340, 341, Dyadic product, 19, 41 476, 484–486, 504, 505, 519, 521, 524 Dynamic boundary condition, 56, 59, 386, strong, 462, 473, 475, 476, 484, 485, 502, 398, 402 504–507, 519, 521, 523–526, 531 Dyson, F.J., 256 weak, 328, 521, 524, 527, 531 Discrete E points, 9, 14, 15 Earth, 215, 216, 276, 295, 303–306, 309, set, 539 316, 334, 341, 342, 461–464 spectrum, 424, 470 quake, 295 vortices, 221 Earth Polychromatic Imaging Camera zeros, 539 (EPIC), 464 Discretization, 9, 10, 16, 224, 250, 531 Eddy, 26, 35, 115, 121, 126, 127, 135, 147, Dispersion 148, 152, 161, 218, 251, 252, 254, normal, 281 338 Index 551

or vortex shedding, 135 Euler, L., 5–7, 17, 19, 26, 28, 31, 43, 44, Eigenfunction, 423, 428, 432, 440, 448, 451 48–50, 53, 55, 56, 88, 92, 96, 154, linearly independent, 428 205, 210, 211, 225, 226, 228, 257, Eigenvalue, 27, 61, 420, 428 285, 291, 297, 299, 300, 311, 314, Eigenvector, 27, 61, 62, 423 317, 324, 343, 344, 459–461, 467, Elastic, 1, 218, 276, 457 487–489, 494, 502, 514, 516 Electromagnetism, 6, 35 Euler–Lagrange equation, 50, 53, 55, 56, 88, Electronic excitation, 458 297 Electrostatics, 47, 252 Evaporation, 491 Elementary particle physics, 420 Evers, W., x Ellipsoid, 27 Evolution, 14, 16, 138, 144, 159, 212, 221, Elliptic 234, 235, 250–252, 290, 323, 517, 531 coordinates, 128 Excess energy, 254 function, 100, 181, 195, 197–199, 259 Excitation, 207, 249, 276, 385 Elliptical galaxy, 162 Existence proof, 199, 234, 399, 419, 420, Emission, 458, 466 422, 433, 441, 452–454, 512 Encyclopedia Britannica, ix, 408 Explosion, 9, 323, 473, 474, 477 Energy Exponential growth, 143, 228, 256, 286, 462 balance, 470 Extended complex plane, 98, 99 law of mechanics, 479 Exterior differential, 63, 65, 66, 68, 72 rotation, 458 transfer, 385, 393–395 Enthalpy, 43 F Entire function, 359, 362, 427 Fabula, A.G., 192 Entropy, 209, 213, 216, 254, 336, 338, 458, Factorization, 128, 129, 332, 333, 343, 355, 473, 476, 478, 481, 485, 486, 492, 435 503, 511 Failed wind, 490–492 increase, 336, 338, 476, 485 Feedback, 286 Environment, 216 Feldmeier, A., 291, 292, 304 Equation Fenton, J.D., 399 of motion, 19, 20, 122, 154, 231, 250, Fibration, 35 299, 302, 312, 313, 323, 324, 328, 343, Field theory, 6, 50, 51, 63 345, 368, 372, 383, 397, 402 Filtering, 225 of state, 6, 11 First law of thermodynamics, 479 Equator, 215, 304, 306, 307 First order, 28, 57, 188, 198, 199, 202, 219, Equilibrium, 466, 480 225, 228, 229, 237, 279, 291, 314, Equipartition theorem, 480 318, 323, 328, 332, 351, 357, 363, Erdélyi, A., 158, 353–355, 360 368, 369, 371, 372, 383–385, 387, Ertel, H., 216 391, 444, 447, 459, 496, 511, 512, ESA, 152 519, 520, 542 Escape speed, 491 Fisher, J., 222–224 Estuary, 315–323, 335 Five minute oscillations, 470 head, 316–323 Fixed-point theorem, 433 mouth, 316, 318, 321, 322 Flanders, H., 65, 75 Euclid, 5, 35, 98, 104, 443 Flapping of ﬂags, 218 Euler Flat plate, 115, 128–130, 172–174, 176, 178, equation, 19, 28, 43, 44, 48–50, 56, 154, 181, 182, 184, 185, 188, 191, 193 205, 210, 211, 225, 226, 228, 257, 285, Fließbach, T., 213 291, 299, 300, 311–314, 317, 324, 343, Flow 344, 459–461, 467, 487–489, 494, 502, past step, 164, 165 514, 516 separation, 127, 163, 164, 166, 260 picture, 5–7, 31 Flügge, S., 351 552 Index

Fluid mode, 234, 375 boundary, 8, 12–14, 28, 35, 90, 131, 132, transform, see Transformation, Fourier 148, 161, 163, 164, 166, 172, 174, 192, Fourier–Bessel theorem, 158 201, 203, 297, 368, 369, 386, 412, 470 Fourier, J., 150, 151, 158, 216, 221, 234, 235, parcel, 5–15, 17–20, 24–28, 31, 32, 36, 241, 246, 247, 255, 257, 277, 280, 38, 51, 61, 63, 79, 93, 121, 153–155, 287, 351–354, 356, 357, 359, 360, 207, 209, 211, 213, 216, 231, 235, 275, 371–375, 423, 450 276, 282, 284, 286, 295, 297, 300, 301, Fournier, J.D., 507, 508, 512 304, 306, 334, 336, 341, 370, 371, 396, Fox, M.J.H., ix, 404 399, 459, 464–466, 476, 479, 503, 508, Fraenkel, L.E., 152, 399 526, 527, 529, 530 Frame of reference, 5, 8–10, 32, 132, ring, 155 210, 216, 261, 279–282, 285, 335, tearing and fracture, 5, 12 338–340, 476, 481, 482, 485, 503, Flux, 276, 396, 463, 476, 478, 484, 485, 504, 521, 526 489–491 Fredholm Focus, 502 alternative, 431 Fog, 477 determinant, 427, 431–433 Foliation, 211, 512 equation, 422 Force Fredholm, I., 422, 427, 431–433 apparent, 10, 341, 342 Free-fall law, 301 body or volume, 17 Freeman Sound, 307 buoyancy, 256, 275, 276, 457, 464–467 Free surface (see also: Wave, free sur- complex, 131, 132 face), 56, 57, 59, 296, 297, 303, 328, conservative, 29, 44, 63, 266 367–369, 371–376, 385, 386, 394, contact, 17, 284 398, 399, 402–404, 406, 410, 411, Coriolis, see Coriolis acceleration 414, 420 Coulomb, 276 Freeway, 477 density, 9, 410 Frenet formulas, 263 drag, 132, 153, 172 Frenet, J.F., 261, 263 equilibrium, 17, 225, 276, 303, 306, 343, Frequency shift, 285 344, 464 Fresnel, A.J., 380 external, 9, 31, 209, 275, 284, 334, 367 Fresnel lense, 380 free, 32, 38, 341, 344 Friction, 24, 26–28, 31, 35, 46, 121, 130, friction or viscous, 12, 14, 19, 26, 154, 152–155, 158, 216, 218, 316, 476 155, 284, 362, 363, 476 Friedrichs, K.O., 56, 477, 479, 486, 502, 504, horizontal, 302 512, 514, 519, 525, 526, 528–531 inertial, 256, 275 Frozen-in ﬁeld, 32 internal, 9, 17, 20, 22, 275 Function lift, 115, 130–132, 134 continuous, 12–14, 19, 28, 34, 35, 64, 84, material, 275, 295 85, 89–91, 96, 97, 109–113, 186, 187, nonlocal, 284–286 350, 423, 424, 426–428, 433, 436, 444, on body, 130–132, 172, 479 452, 453, 504, 521, 536–538 radiative, 275, 284–286, 291, 292 element, 358, 405, 543 shear, 17, 20, 21, 24, 25, 362, 363, 386 Functional, 91 short range, 17, 284 action, 213 surface, 17, 18, 20, 24, 284, 408 analysis, 249 surface tension, 410 analysis, nonlinear, 367, 419, 420, 433 tidal, 295, 302–305, 309, 334 Forced solution, 303, 305 Foul stream, 394 G Fourier, 360 Galilei, G., 80, 141, 396 coefﬁcient, 151, 216, 241, 246, 247 Gamelin, T.W., 109 integral, see Transformation, Fourier Garding, L., 88 Index 553

Gas Growth rate, 207, 221, 228 dynamics, 512, 514, 516 Gulf of California, 464 ring, 2, 153–155, 157, 159 Gasoline, 207 Gauss, C.F., 8, 11, 19, 34, 63, 67, 81, 87, 93, H 94, 290–292, 348, 446, 484 Hadamard, J., 88, 249 Gaussian, 290–292 Hagen, J.O., x General relativity, 420 Half Generating function, 54, 238, 241 axis, 351 Genus, 211, 212 plane, 101, 104–106, 182, 187, 193, 195, Geometrical optics, 282 199, 237, 352, 353, 358, 359, 361, 362, Geophysical fluid dynamics, 457 404–406, 441, 511, 543 Gerstner, F.J., 419 unknown, 351, 362 Gilbarg, D., 164, 172, 181, 184, 193 Hama, F.R., 224 Gjevik, B., x, 307 Hamilton equations, 14, 15, 52, 54, 55, 60, Glacier, 27, 362 223, 252 Global Hamiltonian, 43, 50, 52, 57, 60, 223, 224, analytic function, 358 252, 454 constant, 50, 229, 411, 524 Hamilton, W.R., 14, 15, 43, 50, 52, 54, 55, solution, 270 57, 60, 223, 252, 454 Glue, 27 Hankel, H., 158, 348 Goddard Space Flight Center, 117, 310 function, 348 Goldstein, S., 256 transform, 158 Golusin, G.M., 98, 174 Harbor, 303 Google Earth, ix, 377 Hargreaves, R., 56 Goursat, E., 535 Harmonic Goursat lemma, 535 function, 81–86, 88, 201, 397, 443, 448 Gradshteyn, I.S., 221 wave, see Wave, harmonic Grant, M.A., 402–404, 406, 407 Havelock, T., 166 Graph, 100 Headform, 163 Grattan-Guinness, I., 96 Heat, 336–338, 458, 474, 476, 477, 479, 482, Gravel, 363 489, 491, 492, 507, 512 Gravitational collapse, 9, 152 capacity, 215 Gravity, 17, 56, 131, 153, 228, 255, 275, 284, conduction, 12, 458, 479, 489 299, 303, 367, 402, 409, 415, 416, equation, 512 457, 479, 487, 489, 502 exchange, 458, 465 wave, see Wave, internal gravity generation, 338, 458 Green, G., 3, 47, 81, 82, 84, 85, 87, 88, 204, transfer, 334 230, 259, 292, 347–349, 368, 376, Heating, 215, 462, 467, 482, 489, 492 441, 443, 445, 447, 448 Heaviside, O., 287 function, 47, 85, 87, 230, 259, 292, Hedstrom, G.W., 321 347–349, 368, 371, 376, 443 Heley Sound, 307 integral theorem, 81, 82, 84, 87, 88, 204, Helioseismology, 3, 470, 471 349, 441, 443, 445, 447, 448 Helix, 136, 138, 261–266 Greenland, 137 Hellinger, E., 422, 424, 427, 433 Greenspan, H.P., 328, 333 Helmholtz, H., 3, 45, 51, 90, 119, 163, Greenwich, 308 167, 171, 172, 207–209, 211, 212, Grid 216–218, 221, 225, 228, 229, 234, compression, 10 248, 251, 252, 255, 257, 259, 266, numerical, see Mesh 347 Griffiths, D.J., 279 decomposition, 51, 90 Gröbli, W., 260 theorem on infinite speed, 119 Grossman, N., 315 vortex theorem, 211, 217 554 Index

Hemisphere, 305, 343 Image force, 252 Henchel islands, 307 Im grossen, in the large, 419, 420 Herglotz, G., 5, 282 Im kleinen, in the small, 201, 419, 420 Hering, A., x, 475 Immersion, 3, 14, 128, 130–132 Hessen, 207 Imperfect nozzle, 491, 492 Hess, S.L., 215 Implosion, 161, 323 Hexagon, 400 Incompressible, 1, 11, 43, 49, 50, 56, 61, 130, Hibberd, S., 334 207, 225, 250, 255, 296, 335, 341, Hicks, W.M., 260 368, 371, 385, 395, 397, 402, 420, Higdon, J.J.L., 250 422, 457, 464, 468, 488 Higher order terms, 246 Inconsistency, 83, 240 High water, 306–309 Inductance, 268 Hilbert, D., 513, 515, 517, 521, 523–525 Induction proof, 69, 70, 75 Hill, E.L., 213 Inertial frame, 32, 132, 216, 335, 338, 396, Ho, C.-M., 252 399, 476 Hölder condition, 233 Infinite Hölder, O., 233 speed, 30, 119, 130, 463 Hole, 12–14, 64, 84, 98, 99, 171, 535, 536 strip, 105, 106, 168, 169, 172, 182, 200, Holland, 308 203 Holomorphic, 98 Inflexion point, 121, 185, 186, 195, 257, 258 Holzer, T.E., xii, 491, 492 Inflow/outflow, 8, 9, 29, 115, 154, 167, 168, Homologous, 536 171, 174, 281, 326, 478, 485, 489 Homotopic, 64, 543 Information, 290, 291 Hopf, E., 351, 355, 356, 361–363, 512 Inhomogeneous equation, 149, 383–385, Hop tendrils, 262 439, 522, 524 Huang, K., 29, 31, 457 Initial values or conditions, 5, 14, 15, 36, Hubble, E.P., 152, 162 153, 157, 158, 234, 235, 237, 239, Hubble Space Telescope, 152, 162 240, 249, 250, 252, 290, 292, 314, Huerre, P., 252 371, 372, 374, 473, 507–510, 513, Hugoniot, P.-H., 485 517–519, 531 Hundhausen, A.J., 33 Injective function, 243, 539 Hurwitz, A., 113, 232, 233 Instability, 3, 121, 207, 216, 258, 260, Hutson, V., 419, 420, 433 284–286, 462, 466, 467 Hutter, K., 311, 341, 344, 351, 362, 363 buoyancy (Schwarzschild), 466, 467 Huygens, C., 282, 378 Kelvin–Helmholtz, 3, 207–209, 216, Huygens principle, 282, 378 218, 221, 225, 228, 229, 234, 248, 251, Hwang, L.-S., 334 252, 255, 257 Hydraulic jump, 3, 295, 334, 335 line-driven, 286, 287, 291, 292 Hydrotechnik Lübeck GmbH, x, 395 Rayleigh–Taylor, 229, 255 Hyperbola, 128, 130, 404, 407, 470, 471 vortex pattern, 138, 143 Hyperbolic Instantaneous, 31, 334, 335, 476–478, 502 coordinates, 128 Integrable, 246, 363 type, 519–521, 523 square, 277 Integral curve, 31, 32, 268, 270, 501 Integral equation, 3, 166, 350, 351, 359, 419, I 422, 424, 426, 427, 429, 433, 438, Ice, 362, 363 440, 441, 443, 448, 449 Ideal homogeneous, 423, 427, 428, 431, 448, fluid, 458 449, 451 gas, 6, 20, 209, 215, 458, 460, 480, 502 inhomogeneous, 431–433, 439 gas law, 480 linear, 422, 432, 433, 439, 448, 449, 451 Identity theorem, 538, 540 nonlinear, 3, 419, 420, 422, 433, 435, Ill-posed, 249 436, 438, 439, 441, 443, 447, 448, 453 Index 555

Urysohn, 419 Jeans, J., 31 Integral kernel, see Kernel Jeffreys, H., 153, 154, 383, 385 Integro-differential equation, 286 Jet, 2, 79, 100, 113, 161–163, 168, 171, 172, Interface, 14, 225, 227–230, 284, 416, 531 193–195, 199–201, 367, 403, 406, Interference, 279, 280, 378, 470 409 Interior, 12, 85, 90, 104, 182, 183, 187, 189, propulsion, 473, 478 349, 441, 470, 535–538, 541–543 Propulsion Laboratory, 310 Interstellar gas, 152, 474, 491 Johnson Space Center, 316 Invariance, 32, 34, 55, 61, 62, 80, 183, 213, Jouguet, E., 486 267, 269, 516, 524, 525 Joukowski, N., 115, 130 of domain theorem, 13, 539 Joyce, G., ix, 254, 255 Inverse function theorem, 315 Jump, see Discontinuity Inversion, 5, 158, 183, 412, 512 condition, 4, 334, 477, 481–483, 485 theorem (Lagrange), 315 Inviscid, 2, 12, 14, 19, 28, 35, 45, 46, 48, 50, 51, 56, 79, 93, 94, 115, 118, 121, 130, K 132, 134, 147, 148, 164, 172, 207, Kabanikhin, S.I., 249 209, 210, 216, 218, 225, 248, 252, KAM theorem, 454 255, 257, 266, 296, 335, 341, 368, Kansas, 212 371, 385, 395, 397, 402, 409, 420, Kant, I., 153 422, 475, 477, 493, 502, 507, 512 Kapoulitsas, G.M., 362 Ion, 1, 29, 30, 32, 275, 276, 284, 285, 291, Kármán, Th., von, 2, 115, 134–136, 138, 292 140, 143, 145, 146 Ionization, 458 Karp, S.N., 350, 351 IR and UV emission, 474 Keady, G., 367 Ireland, 163 Keller, H., 340 Irrotational, 2, 35, 49, 50, 56, 63, 79, 80, Keller, J.B., 415 90–93, 98, 132, 134, 147, 189, 192, Kellogg, O.D., 81, 92, 102 385, 386, 395, 397, 402, 404, 409, Kelvin–Helmholtz theorem, 207, 210–212 420, 421, 493 Kelvin, Lord, 33, 90, 207–209, 211, 212, Isentropic, 209, 216, 476, 478 216, 218, 221, 225, 228, 229, 234, Isobars, 215 248, 251, 252, 255, 257, 266, 311, Isochores, 215 341, 343–347, 352, 362, 476 Isocontour, 93, 94, 152, 216 Kelvin principle, 90, 91 Isothermal, 458, 460 Kennel, C.F., 276 atmosphere, 461, 462, 464–469 Kepler, J., 115, 153, 156 ﬂow, 209, 489, 494, 516 Kernel, 286, 287, 290, 351–355, 357, 359, Isotropic 422, 424, 426, 427, 429, 433, 434, pressure, 17, 411 436, 448, 450–452 turbulence, 61–63 integrable, 419 Isotropy, 213, 348, 385 Kida, S., 37, 250, 512 Iteration, 16, 323, 383, 420, 425, 435, 437, Kinematic, 17, 44, 135, 213, 218, 258, 438, 448 264–266, 275, 281, 286, 297, 326, 344, 392, 397, 473, 476, 477, 483, 486, 507–509, 512, 519 J boundary condition, 56, 59, 163, 166, Jackiw, R., 51 227, 297, 368, 386, 397, 402, 421 Jackson, J.D., 82 of discontinuities, 473, 476, 477, 483, Jacobi, C.G.J., 54, 55, 69, 100, 195, 197, 198, 486 204 of vortex rings, 258 Jänich, K., 19, 83–85, 543 space–time diagram, 326 Jan Mayen island, 136, 137 wave, 281, 286 Jeans equation, 31 Kinetic 556 Index

energy, 90, 91, 252–254, 277, 336, 394, equation, 59, 79–81, 83, 85, 87, 88, 90, 476, 477, 479, 486 91, 97, 119, 133, 369, 389, 390, 397, theory, 1, 15, 28, 458, 477, 480, 491 409, 495 Kinnersley, W., 412, 415 transform, see Transformation, Laplace Kinsman, B., 409 Laplacian, 47, 443 Kirchhoff, G.R., 167, 171, 172, 222, 223, Large-scale, 138, 251, 278, 401, 512 466 Laurent, P.A., 108, 131, 427, 540, 541 Kirchhoff method, 171 Laurent theorem, 541 Knopp, K., 100 Laval, G.de, 473, 474, 486–492, 500–502 Knot, 266–269 Laval nozzle, 473, 474, 486–493, 498, Knottedness, 269, 270 500–502 Kolmogorov, A.N., 454 Lax, P.D., 484, 486, 513 Kolscher, M., 2, 184, 185, 189, 191, 192 LeBlond, P.H., 341, 346 Kolsky, H., xi, 473, 474 Legendre, A.-M., 60 Konrad, J.H., ix, 252 Leibacher, J., 468, 471 Kotschin, N.J., 35, 37, 129, 138, 140, 259, 375, 394, 408 Leighton, R.B., 470 Kraichnan, R.H., 61 Le Lacheur, E.A., 344 Krasnoselskii, M.A., 419, 423, 433, 454 Lemma of Schwarz, 539, 540 Krasny, R., 224, 225, 250 Leppington, F.G., 362 Krasovskii, Yu.P., 3, 367, 420, 454 Levandosky, J., 446 Krehl, P.O., 477, 512 Levi-Civita, T., 2, 25, 182, 195, 420, 422, Kronauer, R.E., xi, 224 441 Kutta–Joukowski theorem, 115, 130 method, 182, 195 Kutta, M.W., 115, 130, 134 symbol, 25 Lewy, H., 519 Lichtenstein, L., 3, 12, 13, 420, 429, 433, L 435, 436, 439–441, 443, 448, 452, Lagrange, J.-L., 5–7, 10, 11, 36, 44, 50, 453 53–57, 59, 60, 88, 209, 227, 231, 235, Lifshitz, E.M., 119, 213, 458, 482, 486, 503, 265, 297, 315 511, 526 coordinate, 231, 235 Light derivative, 6, 7, 10, 11, 209, 227, 265 ray, 381 function, 2, 50, 54, 57, 59 speed of, 277, 285 marker, 5, 36 year, 162 picture, 5–7 Lighthill, J., 27, 28, 281, 369 Lagrangian, 56, 60 Limit point, 99, 538, 540 Laitone, E.V., 297, 411 Lake, 303, 308 Lindelöf, E.L., 419 Geneva, 303 Line Texoma, 116 drag effect, 291 Lamb, H., 11, 17, 91, 93, 118, 129, 143, 172, vortex, 32, 34, 35, 37, 46, 135, 138, 207, 210, 218, 220, 221, 259, 277, 278, 216–218, 252, 258, 261, 263–269 280, 281, 298, 304, 305, 307, 312, Linear 346, 362, 375, 383, 462, 476 equation, 125, 148, 255, 332, 383, 384, Laminar, 252 431 Landau, L.D., 119, 213, 458, 482, 486, 503, stability analysis, 144, 216, 219, 221, 511, 526 225–229, 234, 240, 249, 255–258, 265, Landslide, 295 285, 286, 324, 368, 369, 371, 408, 459, Lang, K., 471 461, 467 Laplace, P.-S., 27, 28, 59, 79–81, 83, 85, 87, Linearization, 216, 257, 298, 300, 324, 461, 88, 90, 91, 97, 110, 119, 133, 255, 467 288, 369, 389, 390, 397, 409, 443, Liouville, J., 14, 15, 198, 353, 356, 358, 359, 495 363, 425, 539, 540 Index 557

analytic function theorem, 198, 353, 356, Cox-Clayden, 195 358, 359, 363, 539, 540 ﬂuid motion, 12 iteration method, 425 homotopy, 64, 74, 75, 98, 99, 543 phase space theorem, 14, 15 identity, 64 Liquid, 1, 323, 457, 459 induced, 68, 72 Littman, W., 420, 435 Kirchhoff, 171, 172 Liu, X., 399 Kolscher, 185, 189–191 Localized, 1, 290–292 Levi-Civita, 182, 183, 185, 193 Locally uniformized, 110 linear, 183 London Mathematical Society, x linear complex, 182–184 Longuet-Higgins, M.S., ix, 377, 383, 385, open, 539 390–394, 396, 399, 400, 404, 414, open and closed path, 64, 99, 535 415 pullback, 68–71 Loop, 195, 196, 209, 211, 212, 216, 267 Riemann sheet to complex plane, 109 Los Angeles, 396 Schmieden, 185 Lo, S.H., 218 Schwarz–Christoffel, 100, 101, Love, A.E., 20 104–107, 166, 168, 169, 174, 182, Lucy, L.B., 286, 291 198 Lu, J.-K., 99 to higher forms, 65 Luke, J.C., 56, 57, 59, 60 Weinstein, 200 Lunar semidiurnal tide, 310 Marginal stability, 256, 291 Lushnikov, P.M., 406, 407 Mariana Trench, 295 Lüst, R., 157 Mariotte, E., 458 Lynden-Bell, D., 153, 154, 157, 158 Markushevich, A.I., 13, 102, 103, 107, 235, 359, 539, 543 Mars, 117 M Reconnaissance Orbiter, 117 MacAyeal, D.R., 363 Marsden, J.E., 35, 43, 130, 212, 512, 513 Macchetto, F.D., 162 Mass loss, 491, 492 Mach, E., 334, 473, 475, 478, 482, 483, 488, Material line or surface, 63, 211, 212 492 Maximum property, 79, 83–85, 539 cone, 334, 473, 475, 478 Max-Planck-Institute, 152 number, 482, 483, 486, 488, 492 Maxwell, J.C., 6, 31, 480, 491 Magnetic McCaughrean, M., x, 152 ﬁeld, 32, 35, 36, 343, 470 McCowan, J., 296, 299, 314 induction, 48 McIntyre, M.E., 216 reconnection, 502 Mean free path, 475, 491 Magnetorotational instability, 153 Mean value, 62, 83–85, 232, 233, 389, 434, Magnetostatics, 48 435 Main normal, 262 property, 83–85 Manhattan beach, 396 theorem, 434, 435 Manton, M.J., 362 Meleshko V.V., 159 Map Meridian, 304, 307 1-1, 5, 12–14, 33, 36, 43, 98, 99, 108, Meromorphic function, 79, 99, 108, 113, 109, 112, 113, 166, 174, 231, 235, 281, 535, 540 400, 403, 537 Mesh, 39, 93, 321 analytic, 99, 165, 166, 174, 195 Messier 87, 162 between topological spaces, 64 Meteorology, 216 boundary to boundary, 13 Method of characteristics, 4, 284, 313, 477, closed to exact forms, 73 512, 513, 521, 524, 525, 531 conformal, 79, 100, 101, 105, 108, 134, Metrical factor, 39 166, 174, 193–195, 198, 200, 400, 441, Mexico, 377 539 Meyer, R., 12, 14, 341 558 Index

Meyer, Th., 500, 501 Navier–Stokes equation, 27, 28, 121, 122, Meyer, W., 92 155 Michell, J.H., 399 Nehari, Z., 102, 105, 107, 174, 183 Microstate, 254 Neighborhood, 98, 99, 110, 199, 201, 270, Mihalas, D., 284, 470 280, 290, 348, 404, 537–540, 542 Mihm, S., x, 208 Neighboring, see Adjacent Miles, J.W., 60, 92 Nekrasov, A.I., 419, 420 Milne, E.A., 286 Neumann, C., 82, 83, 90, 91, 109–111, 113, Milne-Thomson, L.M., 9, 17, 18, 24, 26, 37, 124, 133, 362, 363, 411, 424, 425, 50, 129, 133, 172, 277 443, 444, 540 Mineola, 212 Nevanlinna, R., 110 Mixing, 207 Newman, J.N., 166 layer, 252 Newton, I., 6, 24, 27, 31, 154, 285, 303, 465, Mode, 207, 216, 234, 375 476, 479 coupling, 216 Newton’s second law, 285, 465, 479 Modulation, 278, 279, 380 New Zealand, 308 Moffatt, H.K., ix, 127, 266–268 Nirenberg, L., 420 Moment of inertia, 458 Noble, B., 351, 361 Momentum equation, 9, 10, 336 Node, 502 Monatomic gas, 24, 482, 527 Noether, E., 213 Monge, G., 512 Noether theorem, 213 Monna, A.F., 92 Noise, 291, 292 Monodromy theorem, 405, 543 Noll, W., 42 Monomial, 65, 66, 68–70, 72 Non- Montagnon, P.E., 125 inertial frame, 32 Montgomery, D., ix, 254, 255 orthogonal coordinates, 39 Moon, 295, 303–308, 334, 463, 464 uniform ﬂow, 385–389 Moore, D.W., 3, 36, 224, 234, 238–240, Nonlinear 246–250, 401, 507 Morera, G., 537 dynamics, 420 Morera theorem, 537 equation, 239, 313–315, 433, 436, 439, Mortimer, C.H., 341, 344 441, 442, 448, 451 Morton, K.W., 249 interaction, 395 Moser, J., 454 operator, 433 Moving with problem, 235, 249, 409 bore, 335 solution, 3, 334, 367, 419 ﬂuid, 15, 16, 32, 61, 207, 209, 211, 212, stages, 221 266, 285, 459, 526 terms, 144, 234, 239, 240, 249, 285, 292, grid, 9, 10 315, 316 shock, 485, 503 tide, 295, 316 wave, 279–282, 521, 526 wave, see Wave, nonlinear M2 tidal constituent, 310 Nonlocal, see Force, nonlocal Multivalued, 108–110, 112, 243, 507, 508, Norbury, J., 367 511, 514 Nordaustlandet, 307 Muncaster, R.G., 24 Norm ., 249, 419, 434, 437, 440, 452 Mustonen, E., 396 Norman, M., 10, 42, 43 Mysak, L.A., 341, 346 North America, 464 Pole, 95, 341 N Sea, 309, 346 NASA, x, 117, 136, 137, 152, 162, 209, 316, Norway, 362 464 Nozzle throat, 491, 498–500 Navier, C.L.M.H., 27, 28, 121, 155 Null-homolog, 542, 543 Index 559

Numerical calculation, 9, 127, 166, 181, 221, Orifice, 161–163, 167, 168, 170, 174, 224, 225, 250, 255, 291, 292, 303, 199–201 321, 334, 402, 404, 407, 420, 500, Orion Nebula, 152 505, 512 Ormhullet, 307 Orszag, S.A., 61 Orthogonal coordinates, 1, 8, 37, 39, 40, 42 O Orthogonality, 150, 236, 450 Ocean, 83, 136, 216, 295, 303, 305, 306, 308, Oscillation, 3, 136, 209, 275, 276, 278, 279, 309, 328, 334, 341, 344, 347, 362, 291, 295, 306, 343, 344, 362, 375, 367, 394, 457, 464 396, 463–465, 470 Oceanography, 341, 408 Osculating plane, 262, 264 O’Dell, C.R., 152 Osgood, W.F., 108, 539 Offshore, 344, 395 Owocki, S.P., 285, 286, 288–292 Oil, 225 Oklahoma, 116 Olunloyo, V.O.S., 351, 362, 363 P One- Pacific, 308, 309 dimensional, 1-D, 46, 276, 277, 285, 300, Page, M.A., 249, 250 303, 304, 316, 323, 324, 376, 385, 459, Painlevé, P., 407 460, 464, 478, 485, 486, 489, 492, 502, Panofsky, W.K.H., 165 507, 512, 514, 531 Paradox, 132, 172, 290, 507 parameter family, 213, 415, 509, 512, Parallelepiped, 11 513, 516, 517, 521, 525–529 Parallel flow, 129, 184, 194, 218 O’Neill, M.E., 122, 127 Parameterization, 34, 51, 53, 191, 232, 270 Onsager, L., 252, 254 Parker, E.N., 489–492 Open Passats, 215 ball, 83–85, 446, 536 Pathline, 31, 32, 93 cover, 212 Pathological, 3, 234, 250, 259, 286, 291, 292 disk, 86, 87, 100, 405, 443, 444, 446, 447, Peregrine, D.H., 334 536, 539, 540, 543 Perlman, E.S., 162 mapping theorem, 539 Permutation, 25, 27, 40, 66, 263 set or domain, 8, 12–14, 34, 65, 68, 84, Perturbation, 2, 205, 216–221, 224–229, 85, 90, 91, 98–100, 445, 539 234, 250, 256, 257, 280, 284–286, Operator, 419, 422–425, 428, 439 290–292, 316, 317, 324, 329, 343, adjoint, 422 347, 348, 368, 369, 371, 372, compact, 422, 433 459–462, 464–466, 531 contracting, 435 expansion, 316, 317, 385, 387, 388, 391, divergence, 11, 93 392, 399, 459, 467 equation, 424–426, 429, 430, 433, 439 interface, 225, 226, 228, 229, 234, 235, homotopy, 73 237, 247–249, 256 linear, 351, 390, 423, 424, 426, 428, point-like, 348, 368, 371 431–433, 437–439, 449, 451 theory, 317, 319, 321–323 nonlinear, 419, 420, 433, 439, 448 vortex, 36, 138, 140, 142, 143, 146, norm, 434 216–222, 224, 225, 250, 265, 266 product, 424 Petitcodiac River, 335 time evolution, 16, 17 Phase transposed, 422, 423 front, 380 Order (as opposed to chaos, etc.), 1, 224, 254, lag, 276, 307, 310 291 transition, 486 Ordinary differential equation, 120, 220, Phase space, 1, 14, 15, 28, 29, 252, 253, 454 242, 250, 323, 328–330, 332, 373, fluid, 14, 15 496, 501, 503, 505, 512, 515, 519 speed, 15, 29 Orellana, O.F., 239, 246, 248, 249 torus, 454 560 Index

Phillips, M., 165 complex, 2, 81, 96, 97, 115–118, 128, Phillips, O., 395 132, 133, 165–179, 182, 184, 185, Photon, 275, 284, 285, 291, 292 187–189, 191, 193–195, 197, 199–204, flux, 285, 286 367, 397, 399, 402, 403, 405–407, 409, Photosphere, 491, 492 414, 415, 420, 421, 441, 442, 493 Picard, É., 419, 449, 540 dipole, 444, 445 Picard-Lindelöf theorem, 419 energy, 229, 277, 336, 394, 479 Picard theorem, 540 force, 44, 47, 209, 266 Piecewise interaction, 223, 387, 388, 390 analytic, 249 logarithmic, 85 differentiable, 6, 8, 34, 90, 99, 334 theory, 79, 81, 444, 445 Piercing of area by curve, 267 tidal, 304–306 Pillar, cable, 136 vector, 48 Pillbox volume, 478 velocity, 2, 46, 56, 60, 63, 79–81, 86, 88, Piston, 3, 334, 474, 478, 479, 526, 527, 529, 96, 332, 367, 368, 370, 372, 386–388, 530 397, 402, 403, 414, 493 vorticity, 216 Planetary system, 152, 275 Pozrikidis, C., 250 Plasma, 1, 29, 32, 35, 152, 153, 276, 284, Prager, W., 24 285, 457, 473, 489 Prandtl, L., 28 Plemelj, J., 232, 233, 237 Precursor, 334 Plemelj theorem, 232, 233, 237 Pressure, 1, 11, 17, 20, 26, 43, 51, 163, 178, Pneumatic barrier, 394 284, 285, 362, 363, 368, 371, 374, Poincaré, H., 2, 63–66, 71, 75, 80, 90, 341, 421 344–347 as Lagrange function, 1, 50, 54, 57 Poincaré lemma, 63, 66 at contact discontinuity, 505 converse, 63–65, 71, 75, 80, 90 continuous at interfaces, 228, 229 Point from surface tension, 410 at infinity, 98, 99, 104–106, 109, 110, hydrostatic, 229, 276, 299, 336, 344, 409, 178, 194, 195 460, 464, 465 charge, 47, 234, 276 in free surface, 299, 368, 371, 374, 386 mass, 5, 6, 156, 159, 234, 341, 458 negative, 161 mechanics, 5, 10, 39, 50, 54, 55 perturbation, 226, 462, 465, 467 source, 283 p(ρ) alone, 6, 43, 48, 209, 213, 459, 494, vortex, 224, 225, 230, 252 516 Poiseuille flow, 362 postshock, 473, 481–483, 485, 505–507 Poiseuille, J.L.M., 362 preshock, 505 Poisson equation, 46, 47, 147, 230, 259 ram, 462 Poisson, S.D., 46, 47, 147, 230, 259, 371 scalar, 6, 411 Polar coordinates, 45, 86, 98, 118, 119, 122, thermal, 1, 6, 153, 154, 229, 275, 457, 153, 189, 204, 341, 348, 396, 397, 458, 462, 473, 478, 481, 482, 485, 486, 400, 441, 443 488, 489, 491, 504, 505, 507, 512, 526 Pole, 112, 187–190, 197–199, 237, 401, 427, work, 336, 479 507, 512, 540, 542 Price, A., x, 335 condensation, 512 Principal isolated, 540, 542 axes, 27 Poles on Earth, 215, 307 value, 232, 233, 445 Polish circle, 64 Pringle, J.E., 153, 154, 157, 158 Polygon, 101–104, 106, 164, 166, 169, Prism, 381 172–174, 178, 182, 185 Projectile, 334, 473, 475 Polynomial, 359, 403, 497, 510, 540 Projection, 74, 75 Polytropic, 6, 485, 486, 492 Propeller, 161 Potential Prosperetti, A., 47 Index 561

Protoplanetary disk, 152 Rest frame, 335 Proudman, J., x, 304, 309, 312, 314–316, Retardation, 3, 323 318, 319, 321–323, 345 Reversible/irreversible, 12, 61 Pullin, D.I., 233 Reynolds, O., 135, 277 Pulvirenti, M., 35 Rheology, 27 Pym, J.S., 419, 420, 433 Riabouchinsky, D., 178, 179, 181 Rice University, 152 Richardson, L.F., 256 Q Richardson number, 256 Qualitative theory, 420 Richtmyer, R.D., 249 Quantum theory, 50, 458 Riemann Quasi-linear, 313, 515, 519 continuation theorem, 538 invariant, 512, 516, 524, 525 R mapping theorem, 100, 101, 174 Radiation, 284–286, 291, 458, 466, 470, 482 problem, 530 hydrodynamics, 284 sheet, 108–113, 193–196, 199, 244, 407, stress tensor, 394, 395 507, 510, 511 Radiative solver, 531 cooling, 334 surface, 2, 79, 108, 109, 111–113, 195, equilibrium, 466 196, 407, 507, 511 Radiator, 467 Riemann, B., 2, 44, 79, 81, 96–98, 100, 101, Raizer, Yu.P., 323, 458, 477, 478, 486 108, 109, 111–113, 116, 133, 165, Random 168, 174, 185, 193–196, 199, 203, distribution, 279 204, 220, 244, 313, 407, 409, 422, variables, 62 426, 442, 443, 448, 507, 510–513, velocity field, 61 515, 516, 524, 525, 530, 531, 538 Rankine, W.J., 94, 407, 485 Rigid body rotation, 26–28, 155, 470 Rate of change, 8, 15, 211 RiodelaPlata,315, 316 Rational function, 113, 540 Ripples, 408 Rayleigh inflexion theorem, 257, 258 River, 295, 296, 304, 306, 307, 315, 316, 334 Rayleigh, Lord, 123, 229, 255, 257, 258, delta, 316 338, 476 Rock, 362, 363 Ray, R., 310 Roll-up, 217, 218, 221, 224, 225, 234, 247, Re- 250 emission, 291, 292 Rosenhead, L., x, 25, 221, 222, 224, 225 entrant jet, 2, 113, 193, 194 Roshko, A., 255 Real analytic, 235, 246 Rossby, C.-G., 216, 344 Rectifiable, 99 Rossby radius, 344 Recursively, 238 Reflection principle, 185–187, 405 Rotary current, 311, 344–346 Region of influence, 518 Rotation, 3, 15, 24, 26–28, 31, 32, 62, 85, 99, Regular, 34, 358 101, 103, 116, 127, 152, 154, 155, Regularity condition, 191, 490 197, 216, 224, 250, 265, 303, 304, Reid, W.H., 216, 257 306, 309, 334, 341, 342, 344, 346, Relative 353, 354, 357, 458, 470 compact, 433 Rott, N., 230, 233, 247, 250 topology, 84 Rouche, E., 543 Remmert, R., 99, 535–543 Rouche theorem, 543 Residue, 189, 198, 199, 354, 361, 542 Royal Society of London, x theorem, 189, 198, 354, 542 Rundoff errors, 225 Resistance, 161, 172 Run-up, 334, 341 Resolvent, 424–426, 431, 432, 439–441, 451 Rybicki, G.B., 285, 286, 288–291 Resonance, 136, 362 Ryzhik, I.M., 221 562 Index

S Second order, 28, 83, 143–145, 263, 286, Saddle, 502 316, 318, 319, 321–323, 332, 341, Saffman, P.G., 34, 35, 142, 224, 233, 234, 369, 383, 385, 434, 437, 444, 448, 241, 255 519–521, 523 Sagdeev, R.Z., 276 perturbations, 2, 143–145, 202, 227, 276, Sakai, S., x, 469 316, 318, 319, 321–323, 369, 385–388, Salinity, 457 391, 392, 395 San Juanico, 377 Section in Poincaré lemma, 72 Satellite, 136, 137 Secular evolution, 157 Saturn, 207, 209 Sedov, L.I., 323 Savart, F., 48 Semicircle, 182, 183, 187, 190, 193, 195, Scalar 237, 444 field, 5–7, 9, 10, 17, 20, 43, 46, 48, 63, Semi-infinite, 105, 106, 169, 256, 312, 346, 79–81, 88, 90, 213, 216, 230, 249, 270, 347, 351 283, 325, 328, 341, 464, 479 Series, 108, 135, 399, 426, 535 triple product, 11, 264, 265 double, 434, 538 Scale height, 256, 461, 462, 464 Fourier, 150, 151, 221, 235, 450 Scattering, 275, 291, 292, 473 geometric, 236, 239, 450 Schaefer, H., 24 Hadamard, 88, 89 Schauder fixed-point theorem, 433 Laurent, 108, 131, 427, 540, 541 Levi-Civita, 420 Schauder, J., 433 Neumann, 424, 425 Schlieren technique, 473–475 normal convergence, 537, 541 Schmidt power, 98, 108, 312, 314, 315, 317, 358, integral equation, 433, 436 375, 406, 408, 426, 449 orthogonalization, 428 Taylor, 98, 144, 202, 237, 279, 289, 357, theorem, 452 358, 369, 386, 391, 433–435, 437, 449, Schmidt, E., 420, 428, 433, 436, 439, 446, 495, 537, 538 448, 452, 453 uniform convergence, 436, 438, 448, Schmieden, C., 140, 143, 167, 185, 186, 192 535, 538 Schmitz, H.P., 362 Weierstrass, 540 Schoenflies, A., 539 Serrin, J., 92, 476 Schroeter, D.F., 279 Shear flow, 20, 24–28, 36, 61, 154, 207, 218, Schubert, G., 127 221, 225, 229, 230, 234, 248, 250, Schwartz, J.T., 424, 433 251, 255, 257, 362 Schwartz, L., 399, 402–404, 407, 415 Shelley, M.J., ix, 234, 250, 251 Schwarz–Christoffel theorem, 79, 100–102, Shen, M.C., 341 164, 167, 171, 178, 198 Shock Schwarz, H.A., 79, 100–102, 104, 105, 107, forward, 476, 477, 484, 504–506 164, 166–169, 171, 174, 178, 185, frame, 476–479, 481–486, 503, 504 198, 204, 405, 539, 540 front, transition, layer, 2, 3, 205, 334, Schwarzschild criterion, 464, 466 401, 462, 473–486, 491, 492, 501–507, Schwarzschild, K., 464, 466 511, 512, 514, 517, 519, 521, 531 Schwinger, J.S., 351 heating, 474, 476, 482, 507 Screen, 346 interaction, 531 Scruton, C., 136, 138 isothermal, 482 Scruton strake, 136, 138 jump, 473, 476, 477, 482–486, 491, 492, Sea, 303, 304, 307, 309, 315, 316, 341, 377, 507, 511 381, 383, 385, 394 layering, 477 bed, 394 levitation, 462 life, 395 oblique, 478 Second law of thermodynamics, 336, 338, pair, 484, 502, 504–506 485, 492 perpendicular, 478 Index 563

post-, 476–478, 481–486, 504–507 Sobolev norm, 249 pre-, 476–478, 481, 485, 486, 504, 505 Sobolev, S.L., 249 pressure, see Pressure, postshock Sobolev, V.V., 291 rarefaction, 486 Solar reverse, 477, 484, 492, 504–506 atmosphere, 466, 489, 502 speed, 473, 476, 477, 481–486, 503, 505, eclipse, 463 519, 521, 531 flare, 502, 504 speed relative to, 474, 476–479, interior, 470, 471 481–484, 486, 504 plasma, 470 strong, 481, 483, 484, 504, 505 radius, 470 tube, 473, 475, 483, 484, 507 wind, 4, 32, 486, 489–492, 502, 504 weak, 482 Solenoidal, see Divergence free Shore, 328, 329, 333, 334, 338, 340, 341 Solid body, obstacle, 14, 22, 26, 27, 35, 128, Sichel, M., 501, 502 130–132, 135, 136, 161, 163, 166, Sickle, 171, 172 172, 182, 185, 194, 346, 477, 483 Signal speed, 290–292, 326 Solid-body rotation, see Rigid body rotation Silk thread, 218 Soliton, 299 Silverdale, 335 Sollid, J.L., 307 Similarity Solomon, P.M., 286 exponent, 323, 325 Sommerfeld, A., 346, 370 problem, 323 Sonic point, 489, 490, 492, 494–496, 499 solution, 323, 324, 326, 329, 502, 503 Sound, see Wave, sound variable, 323, 325, 502, 503, 505 Sound speed, 285, 457, 459, 460, 463, 464, Simon, M., 502 500, 503, 505, 514, 517, 519, 526, Simple 529, 531 boundary, 34 adiabatic, 458, 459, 461, 465, 482, 487 curve, 98, 541–543 isothermal, 458–460, 464, 494 pole, 188, 197, 199 Source or sink, 8, 86, 115, 148, 476 wave, see Wave, simple South Pole, 95, 341 zero, 199 Spacelike curve, 518 Single-valued, 93, 108, 110, 172, 277, 511, Sparks, W.B., 162 535 Spectral line, 284 Singularity, 3, 34, 36, 87, 95, 134, 173, 225, Spherical coordinates, 39, 62, 94 230–232, 234, 237, 242, 244–251, Spilling breaker, 3, 399 288, 289, 341, 357, 358, 361, 363, Spin, 154, 216, 231, 252, 254, 255 401–408, 443, 445, 452, 490, 507, Spitzbergen, 307 540, 543 Spitzer, M., x, 408 condition, 490 Spreiter, J.R., 484 essential, 540, 542 Sprinkler, 31, 32 isolated, 234, 540 Stability, 115, 138, 140, 145, 146, 221, 225, merging, 402, 407, 512 255–258, 285 order, 401–404, 407 Stagnation point, 129, 133, 172, 174, 176, Slit 193–195 in 3-D, 147, 148, 167, 171 Stallmann, F., 189 in complex plane, 108, 174, 184, 194, 195 Star, 9, 115, 152, 285, 474, 491 theorem, 174 cluster, 29 Small-scale, 27, 61, 115, 138, 221, 241, 249, Star-shaped, 536 251, 252, 338, 512 State variable, 458 Smirnow, W.I., 422, 423, 425, 427–429, 431 Stationary, 199, 205, 207, 285, 323 Smoke ring, 258 flow, 5, 7, 31, 32, 48, 49, 79, 93, 98, 121, Snell law, 378, 379, 381, 382 122, 130, 132, 134, 147, 153, 172, 192, Snell, R., 378, 379, 381, 382 367, 385, 388, 396–399, 402, 409, 420, Soap film, 218 486, 489, 491, 492 564 Index

helical line, 265 Sturrock, P.A., 484 wave, 396, 397, 399, 402 Subsonic, 420, 476, 488, 490, 491, 499, 500, Statistical, 3, 28, 61, 62, 252, 254, 255 505, 518 Stefan, J., 466 Successive approximations, 383, 384, 436, Stegun, I.A., 156, 288, 289, 348, 349, 439, 440, 452 353–355, 360 Sudden impulse, 371, 372, 374, 376 Stein, R.F., 468, 471 Sulem, P.L., 249 Stellar Summation convention, 23, 25, 27, 40, 51, dynamics, 31 61, 192, 222 wind, 491 Sun, 32, 36, 152, 153, 215, 295, 303, 304, Sternberg, W., 444 334, 463, 470, 471, 486, 489–492, Stewart, R.W., 385, 390–394 502, 504 Stirling formula, 245 convection zone, 470 Stirling, J., 245 internal rotation, 470, 471 Stoker, J.J., 297, 324, 336, 338, 351, 354, radiative interior, 470 356, 358, 369, 370, 372, 376, 420, Supercritical speed, 334 453 Superposition principle, 279 Stokes Supersonic angle, 395, 399–403, 407, 420, 421, 453 flow, 334, 473–478, 483, 485, 486, equation, 122 488–491, 495, 499, 500, 502, 512, 531 hypothesis, 24, 25 pockets, 499–501 Stokes, G.G., 24, 27, 28, 34, 63, 67, 116, speed, 474 121, 122, 155, 267, 277, 367, 368, spot, 492 395–404, 407, 420, 421, 453 turbulence, 512 Stone, J.M., 42, 43 Supremum, 434 Stoneley, R., 378, 380 Surface tension, 368, 401, 408, 410, 411, 414 Storfjorden, 307 Surge, 346, 362 Strain, 25, 27, 28, 61, 234 Symmetric Strangford, 163 flow, 125, 126, 133, 147, 148, 163, 178, Stratification, 2, 216, 229, 255, 457, 460, 201, 237, 259, 319, 348, 397, 398, 400, 464, 466 474, 489, 492, 502 Streamfunction, 80, 92–94, 96, 118, 121, kernel, 451 128, 148, 150, 152, 222, 257, 367, tensor, 20, 23–25, 27, 42, 61, 62 397, 403, 493 wave profile, 406 Streamline, 31, 32, 45, 48–50, 93, 94, Synchronized period, 310 118, 119, 126–129, 131, 133, 134, 147, 148, 152, 161, 163–167, 170, 174, 178, 187, 191, 192, 194, 195, T 199, 217, 398, 401, 402, 404, 407, Tail, 527 410–412, 494, 495, 498–500 Tamada, K., 492, 494–496 free, 121, 161, 163, 164, 166, 173, Tanaka, M., 37 174, 177–179, 182, 184–186, 188, 191, Tanveer, S., 406, 407 193–195, 200–202 Tatsumi, T., 512 split, 174, 194, 195, 227 Taylor, B., 98, 144, 202, 237, 279, 289, 357, Stress, 17, 18, 20, 22, 363, 394, 395, 410 358, 369, 386, 387, 391, 433–435, tensor, 410 437, 495, 537, 538 Stretching, 11, 14, 27, 61, 63, 99, 101, 103, Taylor, G.I., x, 229, 255, 311, 312, 323, 500, 292 501 String, 234 Temperature, 1, 6, 11, 254, 277, 457, 458, Strip of analyticity, 355, 356, 358, 362 460, 461, 463, 466, 467, 470, 473, STScI, 162 474, 482, 485, 489, 494, 505–507 Study, E., 540 Tensor Study theorem, 540 analysis, 42 Index 565

Cartesian representation, 24 Toland, J.F., 420 divergence, 18, 19, 21, 42 Tomotika, S., 492, 494–496 divergence theorem, 8 Topology, topological, xv, 12, 64, 75, 84, ﬁeld, 8, 9 186, 212, 267, 269, 401, 420, 490, force stress, 410 502 Gauss theorem, 19 Tornado, 212 inner product, 18 Toroidal, 258 product, 7, 8 Torque, 20, 22, 154, 155 rank, 1, 7–9, 18, 19, 21, 23, 37, 42, 394 Torsion, 261–265 rate-of-strain, 24, 25, 27 radius, 262 stress, 18, 22, 24, 410 Trailing, 184, 250, 251 trace, 25 Trajectory, 6, 7, 14, 15, 31, 32, 139, 275, velocity gradient, 8, 15, 16, 24, 285 280, 370, 371, 399, 410, 485, 519, Tenuous gas, 491 529, 530 Tetrahedron, 17–19 Transcendentally nonlinear, 403 Texas, 116 Transformation Theory of relativity, 518 canonical, 54, 55 Thermal conformal, 98, 99, 400 energy, 479, 480, 492 Fourier, 150, 151, 255, 257, 277, 280, equilibrium, 480 287, 352–354, 356, 357, 359, 360, motion, 458, 475 372–375, 423 pool, 491 Galilei, 141 Thermodynamics, 6, 11, 209, 336, 338, group, 12 457–459, 479 Laplace, 255 Third order perturbations, 315, 317, 323 Legendre, 60 Thomas, R.N., 491, 492 variable, 165, 166, 213, 318, 330, 399, Thomas, T., 291, 292 403, 409, 517 Thompson, M.J., 471 Transient, 334 Thorade, H., 310, 311 Translation, 15, 101, 104, 187, 213, 458 Three-dimensional, 3-D, 14, 35, 48, 50, 55, Transonic ﬂow, 4, 420, 486, 489–492, 88, 147, 199, 234, 260, 261, 263, 369, 500–502 501, 512 Transposed equation, 423, 427 Threshold value, 499 Trenogin, W.A., 420, 435, 439, 453 Tidal Triad, 61, 62 bulge, 305 Trinh, P.H., 402, 406, 407 current, 307, 309, 311, 344, 347 Truesdell, C., 6, 18, 20, 24, 25, 42 friction, 309 Truncation, 290–292 height, 303, 305, 308, 309, 316, 322, 323 Tsiolkovsky State Museum, 487 node, 309 Tsunami, 295 plant, 161, 163 Tube, 31, 32, 34, 37, 211, 212, 214, 215, range, 309 258, 334, 474, 478, 483, 484, 514, stream, 394 526, 529–531 wave, 295, 304, 306–311, 314–316, 318, Tuck, E.O., 334 320–323, 334, 335, 344 Tu, L.W., 19, 66, 71, 73, 75 Tide, 1, 295, 302–311, 315, 316, 323, 334, Turbulence, 3, 61–63, 115, 136, 207, 251, 335, 344 252, 255, 338, 385, 470, 512 inverted, 306, 307 Turner, J.S., 399 nonlinear, 295, 316 Turning angle, 104, 107, 400 Tiling, 195 Two-dimensional, 2-D, 13, 31, 34, 35, 45, Tilted 49, 56, 63, 79, 85, 88, 92–96, 98, 115, plate, 128, 129, 192 121, 122, 127, 130, 132, 134, 135, wedge, 113, 194, 195, 199 147, 153, 164, 205, 230, 252, 255, Tilt of water surface, 299, 303, 328, 362, 363 257, 258, 303, 308, 311, 335, 367, Time step, 224, 531 383, 395, 402, 420, 422, 443, 467, Toeplitz, O., 422, 424, 427, 433 477, 492, 500–502, 504, 539 566 Index

U Vessel, 161, 168, 175, 201, 203, 204 Ulrich, R.K., 470 Vibration, 136, 138, 458 Underpressure, 2, 374, 416 Villat, H.R.P., 191 Underwater Viscoelastic, 1, 27 explosion, 473, 474 Viscosity, 12, 25, 28, 121, 122, 125, 147, 153, landslide, 295 225, 249, 250, 284, 285, 322, 334, Uniformizing parameter, 109 401, 458, 475–477, 501, 502, 507, Unique(ness), 96, 100, 119, 182, 199, 200, 512 234, 259, 336, 419, 436, 452, 481, bulk, 25 490, 510, 517, 518, 521, 522, 541 dynamic, 24, 26 University of first, 24 Oslo, 307 kinematic, 26, 155 Ottawa, 475 second, 25 Sydney, 473 shear, 24 Unna, P.J.H., 394 volume, 25 Unstable Viscous, 2, 115, 118, 121, 122, 124, 127, flow, 258, 286, 292 148, 154, 502 growth, 138, 140, 143, 205, 207, 216, torque, 155 218, 221, 224, 228, 234, 237, 238, 240, Vitali, G., 538 241, 249, 256–258, 275, 284–286, 292, Vitali theorem, 538 468 Volcano mode, 468 Beerenberg, 136 system, 221 eruption, 295, 463 vortex chain, 140, 218 vortex pattern, 138 von Koch, H., 427 wavelengths, 234, 256 von Koppenfels, W., 189 Upstream, 286, 290–292, 477 Vortex, 35, 36, 115, 135–141, 145, 159, 207, Upwelling, 386 216–225, 230, 231, 234, 247, 248, Urysohn, P., 419 251–255, 258, 259, 261, 264, 266, U. S. Army Corps of Engineers, xi, 116 268–270 accumulation, 217, 221, 222, 224, 234, 247, 251, 254 cascade, 115, 512 V chain, 134–142, 218, 219, 221 Vacuum, 161, 163, 164 circular, 258, 259 Väisälä, V., 464, 465, 468, 469 drain, 217 Vallis, G.K., 457 Valve, 161 filament, 34 Vanden-Broeck, J.-M., 415 helical, 261, 263, 265, 266 Vapor, 161, 163 induced speed, 135–140, 217, 219, 222, Variable change, see Transforma- 223, 234, 250, 259, 261, 263, 264 tion,variable interaction, 223, 234, 252 Variational (principle, problem, calculus), knotted and linked, 266–270 50–60, 88, 89, 110, 203, 212 layer, 248, 250, 251, 401 Variation of the line, 32–35, 211, 230, 263, 267, 268 argument, 110 merging, 253, 254 constant, 149 pair, 221, 224, 259–261 Vector ring, 2, 35, 258–261, 263, 266–270 field, 5–7, 9–11, 14, 25, 35, 36, 46, 51, sheet, 35–37, 46, 216–218, 221, 224, 63, 80, 90, 91, 93, 94, 213, 266, 270, 225, 230–235, 243, 244, 246–251, 257, 325, 328, 341, 343, 345 362, 401, 507 gradient, 7, 8, 15, 16, 27, 37, 39, 40, 285 sheet, cylindrical, 37 space, 5, 65, 68, 75, 79, 96, 419 small-scale, 115, 252, 338, 512 Vertical acceleration in water wave, 299, 301 speed, 140, 259, 263 Index 567

straight, 115–117, 133, 134, 138, 140, capillary, 3, 275, 368, 401, 408, 409, 216, 218, 258, 263, 266 414–416, 419 street, 46, 115, 134–136, 138, 140, 146, cavity, 470 221, 224, 225 centered rarefaction, 526–531 strength, 34, 35, 37, 127, 212, 234, 266, centered simple, 525, 526, 529–531 267, 269 cnoidal, 419 torsion, 261, 263–265 collimation, 380, 381, 383 trajectory, 139, 224 Crapper, 3, 368, 408, 409 tube, 32–35, 37, 46, 207, 209, 211, 212, crest, 3, 275, 280–282, 328, 367, 378, 258, 267–269 380, 395–401, 403, 404, 406–408, 414, Vorticity, 25, 27, 32, 34–37, 44–46, 50, 61, 421 94, 95, 115, 117, 130, 147, 152, 211, cylindrical, 341 216, 217, 221, 224, 230, 231, 233, decay, 346, 469 234, 248, 250, 251, 258, 259, 263, destruction, 385, 394 266–268, 270 diffraction, 346, 377 absolute, 216 dispersion, 60, 207, 228, 265, 276, 277, equation, 43, 45 279, 281, 282, 286, 292, 295, 344–347, 368, 370, 371, 377, 378, 408, 462, 463, 467–469, 471 W distortion, 312 Wagner, J., 136 electromagnetic, 343 Wainberg, M.M., 420, 435, 439, 453 energy, 277, 336, 394, 395, 461–463, 467 Wake, 14, 79, 100, 161–163, 172, 173, 177, envelope, 278–281 178, 182, 194, 195, 199, 367, 403, equation, see Waveequation 406, 409 evanescent, 276, 347, 463, 469, 470 Walker, J.D.A., 260 existence proof for nonlinear waves, 419, Wall (as fluid boundary), 28, 118, 120, 123, 420, 422, 433, 436, 441, 454 124, 126, 127, 147, 148, 150, 167, filter, 281 168, 203, 223, 258–260, 326, 328, finite amplitude, 395, 408, 409 363, 398, 474, 477, 483, 500, 507 forced, 303, 305 Water form and profile, 3, 277, 281, 302, 312, bed/ground, 56, 57, 59, 60, 83, 295–298, 314, 318, 328, 333, 344, 367, 395–402, 328, 329, 333, 336, 370, 399 406, 408, 414, 415, 420, 421, 441, 443, column, 295, 464 447 dome, 473, 474 forward facing, 526 sheet, 341 frame, 279–282, 335, 338, 396, 399, 481, weight, 295, 303 482, 485, 503, 504, 521, 526 Wave free surface, 56, 59, 60, 275–278, 296, along vortex, 265, 266 297, 299, 333, 367, 368, 370, 371, amplitude, 207, 217, 276–279, 292, 310, 374–376, 378, 381, 382, 386, 387, 389, 321, 322, 333, 341, 345, 346, 362, 375, 391–396, 398, 400–404, 406–409, 414, 380, 381, 383, 385, 391, 392, 394, 395, 415, 419–422, 441–443, 452, 453, 457, 402, 404, 407, 460, 462, 463, 519 464, 473 and weak discontinuity, 521 front, 275, 282–284, 292, 295, 377, 378, at coast/shore, 295, 328, 329, 333, 334, 380, 385, 523, 526 340, 341, 343, 344, 346, 377, 378, generation, 385 380–383, 385 Gerstner, 419 backward facing, 526 gravo-acoustic, 275, 464, 467–469, 471 barrier, 85, 346, 347, 349–352, 359, 361, group, 275, 277, 278, 280, 281, 380, 470 362 group speed, 3, 275, 277–282, 292, 301, breaking, 295, 299, 328, 332–334, 367, 380, 382, 393, 460, 463, 470 377, 385, 394–396, 399, 402, 407 growth rate, 205, 207, 234, 256, 462 buoyancy, 3, 275, 276, 457, 465, 470 guide, 341 568 Index

harmonic, 207, 275–282, 308, 345–347, 370, 371, 376, 378, 380–382, 386, 389, 371, 375, 377, 378, 380, 384, 390, 394, 392–394, 396, 460, 463, 470 460, 461 plane, 286, 290, 302, 310, 341, 344–347, head, 526, 527 355, 377, 378, 380, 381, 383, 384, 459, height/elevation, 56, 57, 60, 277, 295, 467, 470 296, 298–300, 303, 305, 307–312, 316, Poincaré, 341, 344–347 322–324, 326–329, 332–336, 338, 341, propagation direction, 314, 328, 329, 344, 346, 347, 351, 361, 368, 369, 371, 333, 343–347, 368, 377, 378, 380–383, 372, 374–377, 386, 388–391, 394, 395, 385, 390, 396, 460, 469, 470 403, 404, 415 radiative, 275, 286, 287, 292 highest, 399, 401, 402, 404, 407, 415 rarefaction, 326, 526 in a canal, 295–297, 300, 303, 304, 306, reflection, 3, 311, 321, 333, 341, 470 307, 312, 324, 335, 368, 371, 376 refraction, 377, 379–381, 383, 385, 395 in a stratified medium, 229, 255, 460, 464 rotary, 311, 344–346 in a stream, 385, 386 seismic, 295 incident, 321, 346, 347 shallow water, 275, 276, 295, 298–302, incoming, 314, 333, 346, 347, 355, 356, 304, 306, 312, 314, 316, 318, 321, 323, 362, 382, 394 324, 326–328, 333, 335, 336, 340, 341, interaction with flow, 385, 387, 388, 390, 344–347, 362, 367, 368, 370, 381–384, 394, 395 457, 464 interaction with waves, 279, 395 short, 234, 238, 249, 281, 368, 375, 376, interaction with wind, 383, 394, 414 395 interference, 279, 280, 378, 470 simple, 325, 326, 524–526 internal gravity, 2, 3, 276, 457, 464, sound, 2, 3, 275, 285, 346, 395, 457–465, 468–471 467–471, 500, 503, 505, 514, 519, 521, Kelvin, 311, 341, 343–347, 352, 362 526 kinematic, 281, 344, 392 source, 281, 348, 368 length, 216, 221, 224, 228, 229, 234, 266, spectrum, 234, 470 275, 277–282, 285, 295, 299, 301, 306, speed, 207, 277, 278, 301, 309, 312, 318, 310, 367, 368, 370, 371, 376, 377, 380, 322, 324, 326–328, 334, 339, 396, 399, 381, 385, 386, 394, 395, 400, 401, 414, 402, 441, 521, 526 441, 463 spherical, 283, 378 linear, 3, 276, 295, 298–303, 311, 314, standing, 310, 333, 407, 463, 469, 470 339–341, 343, 368, 369, 383, 384, 386, steepest, 367, 395, 396, 400, 401 396, 404, 451, 459, 461, 462, 467, 521 Stokes, 3, 367, 368, 395, 397, 399–404, long, 229, 281, 370, 375, 376, 395 407 longitudinal, 343, 344, 346 stretching, 292 mode, 205, 216, 284, 468 superposition, 216, 277, 279, 280, 310, node, 278, 281, 310, 311 311, 378, 380, 382, 531 nonlinear, 295, 298, 315, 316, 367, 368, tail, 526, 527 395–397, 399, 401, 403, 408, 419, 420, Tidal, see Tidal wave 422, 433, 436, 441, 442, 448, 452, 453 transmission, 470 number, 157, 207, 228, 256, 276–279, trapping, 470 282, 292, 347, 352, 355, 356, 375, trough, 275 379–382, 385, 389–392, 399, 401, 412, water (general), 3, 56, 60, 85, 92, 414, 469–471, 507 276–278, 334, 377, 378, 380–383, 419, outgoing, 350 420, 422, 433, 445, 448, 453, 457, 521 packet, 278–280, 368, 371 Wave equation, 276, 277, 282, 284, 300, 314 parcel speed in, 301, 526 forced, 303 period, 310, 311, 344, 377, 381, 414 gravo-acoustic, 467 phase, 280, 344 helical wave, 265 phase speed, 3, 209, 275–278, 281, 282, shallow water, 3, 300, 301, 319, 320, 284, 292, 295, 301, 305, 312, 317, 328, 345–347, 383, 384 Index 569

sound, 457, 460, 462 park, 395 surface wave, 56, 59, 60, 372 tunnel, 474, 491 tides, 302, 316 Winding number, 266, 267, 536, 541 Weakly nonlinear, 419, 420, 433, 436, 441, Wine tendrils, 262 442, 452, 453 Winkler, K.-H., 10 Weatherburn, C.E., 19, 24, 40 WKB approximation, 285 Weber function, 348 Wolfson, R.L.T., xi, 491, 492 Weber, H.F., 348 Wormhole, 307 Wedge product, 65, 67 Wright, M.C.M., 47 Wehausen, J.V., 297, 411 Weibel-Mihalas, B., 284, 470 Weierstrass, K., 453, 538, 540 X Weingart, J., 212 X-type topology, 490 Weinstein, A., 199–201, 204 Weinstein’s theorem, 199–201 Well-posed, 234, 249 Y Weyl, H., 110, 486 Yih, C.-S., 2, 147, 148, 152 Whewell, W., 304, 309 YouTube, 116, 208, 260, 464 Whirlpool, 116 Whitening, 473 Whitham, G.B., 60, 281, 392, 394, 477 Z Wiener, N., 3, 351, 355, 356, 361–363 Zabusky, N.J., 250 Wiener–Hopf method, 3, 351, 355, 356, Zarantonello, E.H., 161, 164, 172, 193 361–363 Zel’dovich, Ya.B., 323, 458, 477, 478, 486 Wikipedia, 162, 163, 212, 277, 316, 396, Zemplén, G.V., 476, 486 475, 487 Zenith angle, 304 Williams, J.M., 402 Zermelo, E., 93–95 Williams, W., 362 Zermelo theorem, 94 Wilton, J.R., 408 Zeta function, 220 Wind, 32, 136, 215, 218, 255–257, 285, 367, Zhang, J., x, 218 383, 385, 395, 414, 486, 489, 502 Zierep, J., 489